THE

MATHS
BOOK
 BIG IDEAS
THE ART BOOK THE MATHS BOOK
THE ASTRONOMY BOOK THE MOVIE BOOK
THE BIBLE BOOK THE MYTHOLOGY BOOK
THE BUSINESS BOOK THE PHILOSOPHY BOOK
THE CLASSICAL MUSIC BOOK THE POLITICS BOOK
THE CRIME BOOK THE PSYCHOLOGY BOOK
THE ECOLOGY BOOK THE RELIGIONS BOOK
THE ECONOMICS BOOK THE SCIENCE BOOK
THE FEMINISM BOOK THE SHAKESPEARE BOOK
THE HISTORY BOOK THE SHERLOCK HOLMES BOOK
THE LITERATURE BOOK THE SOCIOLOGY BOOK

SIMPLY EXPLAINED
 THE

MATHS
BOOK
 DK LONDON DK DELHI TOUCAN BOOKS
SENIOR ART EDITOR PROJECT ART EDITOR EDITORIAL DIRECTOR
Gillian Andrews Pooja Pipil Ellen Dupont
SENIOR EDITORS ART EDITOR SENIOR DESIGNER
Camilla Hallinan, Laura Sandford Mridushmita Bose Thomas Keenes
ILLUSTRATIONS ASSISTANT ART EDITOR SENIOR EDITOR
James Graham Nobina Chakravorty Dorothy Stannard
JACKET EDITOR SENIOR EDITOR EDITORS
Emma Dawson Anita Kakar
John Andrews, Tim Harris, Abigail Mitchell,
JACKET DESIGNER EDITOR Rachel Warren Chadd
Surabhi Wadhwa Aadithyan Mohan
EDITORIAL ASSISTANTS
JACKET DESIGN SENIOR JACKET DESIGNER Christina Fleischer, Isobel Rodel, Gage Rull
DEVELOPMENT MANAGER Suhita Dharamjit
Sophia MTT ADDITIONAL TEXT
JACKETS EDITORIAL COORDINATOR Marcus Weeks
PRODUCER, PRE-PRODUCTION Priyanka Sharma
Andy Hilliard EDITORIAL ADVISORS
SENIOR DTP DESIGNER
Tom Le Bas, Robert Snedden
PRODUCER Harish Aggarwal
Rachel Ng INDEXER
DTP DESIGNERS
MANAGING EDITOR Vijay Khandwal, Anita Yadav Marie Lorrimer
Gareth Jones PROOFREADER
PICTURE RESEARCHER
SENIOR MANAGING ART EDITOR Rituraj Singh Richard Beatty
Lee Griffiths First published in Great Britain in 2019
MANAGING JACKETS EDITOR
ASSOCIATE PUBLISHING DIRECTOR Saloni Singh by Dorling Kindersley Limited,
Liz Wheeler 80 Strand, London, WC2R 0RL
PICTURE RESEARCH MANAGER
ART DIRECTOR Taiyaba Khatoon Copyright © 2019
Dorling Kindersley Limited
Karen Self A Penguin Random House Company
PRE-PRODUCTION MANAGER
DESIGN DIRECTOR Balwant Singh
Philip Ormerod 10 9 8 7 6 5 4 3 2 1
PRODUCTION MANAGER 001–311037–Sept/2019
PUBLISHING DIRECTOR Pankaj Sharma
Jonathan Metcalf All rights reserved. No part of this publication may be
SENIOR MANAGING EDITOR
Rohan Sinha reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical,
MANAGING ART EDITOR photocopying, recording or otherwise, without the prior
Sudakshina Basu written permission of the copyright owner.

A CIP catalogue record for this book is

original styling by available from the British Library.
STUDIO 8 ISBN: 978-0-2413-5036-2

Printed in China

A WORLD OF IDEAS:
SEE ALL THERE IS TO KNOW

www.dk.com
CONTRIBUTORS

KARL WARSI, CONSULTANT EDITOR College in Norfolk, UK, for over 20 years. In 2005–06, he was
made a Gatsby Teacher Fellow for creating the popular mathematics
website Risps. In 2016, he founded the competition Ritangle for
Karl Warsi taught mathematics in UK schools and colleges for students of mathematics.
many years. In 2000, he began publishing books on mathematics,
creating bestselling textbook series for secondary-level students,
both in the UK and worldwide. He is committed to inclusion TOM JACKSON
in education, and the idea that people of all ages learn in
different ways. A writer for 25 years, Tom Jackson has written about 200 non-
fiction books for adults and children and contributed to many
JAN DANGERFIELD more on a wide range of science and technology topics. They
include Numbers: How Counting Changed the World; Everything
is Mathematical, a book series with Marcus du Sautoy; and Help
A lecturer and senior examiner in Further Mathematics, Jan Your Kids with Science with Carol Vorderman.
Dangerfield is also a fellow of the UK’s Chartered Institute of
Educational Assessors and a Fellow of the Royal Statistical Society.
She has been a member of the British Society for the History of MUKUL PATEL
Mathematics for more than 30 years.
Mukul Patel, who studied mathematics at Imperial College, London,
writes and collaborates across many disciplines. He is the author
HEATHER DAVIS of We’ve Got Your Number, a book on mathematics for children,
and film scripts voiced by Tilda Swinton. He has also composed
British author and educator Heather Davis has taught mathematics extensively for contemporary choreographers and designed sound
for 30 years. She has published textbooks for Hodder Education installations for architects. He is currently investigating ethical
and managed publications for the UK’s Association of Teachers issues in AI.
of Mathematics. She presents courses for examination boards
both in the UK and internationally and writes and presents
enrichment activities for students. SUE POPE
A mathematics educator, Sue Pope is a long-standing member
JOHN FARNDON of the Association of Teachers of Mathematics and co-runs
workshops on the history of mathematics in teaching at their
A widely published author of popular books on science and conferences. Published widely, she recently co-edited Enriching
nature, John Farndon has been shortlisted five times for the Mathematics in the Primary Curriculum.
Royal Society’s Young People’s Science Book Prize, among
other awards. He has written around 1,000 books on a range
of subjects, including internationally acclaimed titles such as MATT PARKER, FOREWORD
The Oceans Atlas, Do You Think You’re Clever? and Do Not
Open, and contributed to major books such as Science and Originally a maths teacher from Australia, Matt Parker is a now
Science Year by Year. a stand-up comedian, mathematics communicator, and prominent
maths YouTuber on the Numberphile and Stand-up Maths channels,
where his videos have had over 100 million views. Matt performs
JONNY GRIFFITHS live comedy with Festival of the Spoken Nerd and once calculated
pi live in front of a sold-out Royal Albert Hall. He also presents
After studying mathematics and education at Cambridge television and radio programmes for the BBC and the Discovery
University, the Open University, and the University of East channel, and his 2019 book Humble Pi: A Comedy of Maths Errors
Anglia, Jonny Griffiths taught maths at Paston Sixth Form topped the Sunday Times best-seller chart.
6

CONTENTS
12 INTRODUCTION 44 A real number that is
not rational
Irrational numbers

ANCIENT AND 46 The quickest runner

can never overtake
CLASSICAL the slowest

PERIODS
Zeno’s paradoxes of motion

Their combinations
6000 BCE–500 CE 48
give rise to endless
complexities
The Platonic solids

50 Demonstrative knowledge
must rest on necessary
basic truths 60 Exploring pi is like
Syllogistic logic exploring the Universe
Calculating pi
52 The whole is greater than
the part 66 We separate the numbers
Euclid’s Elements as if by some sieve
Eratosthenes’ sieve
58 Counting without
numbers 68 A geometrical tour
The abacus de force
22 Numerals take Conic sections
their places
Positional numbers 70 The art of measuring
triangles
28 The square as the Trigonometry
highest power
76 Numbers can be
Quadratic equations less than nothing
Negative numbers
32 The accurate reckoning
for inquiring into all 80 The very flower
things of arithmetic
The Rhind papyrus Diophantine equations
34 The sum is the same 82 An incomparable star in
in every direction the firmament of wisdom
Magic squares Hypatia
36 Number is the cause 83 The closest approximation
of gods and daemons of pi for a millennium
Pythagoras Zu Chongzhi
 7

142 Nature uses as little

THE MIDDLE AGES as possible of anything
The problem of maxima
500–1500
144 The fly on the ceiling
88 A fortune subtracted Coordinates
from zero is a debt
Zero 152 A device of marvellous
invention
92 Algebra is a scientific art The area under a cycloid
Algebra
154 Three dimensions made
100 Freeing algebra from the by two
constraints of geometry Projective geometry
The binomial theorem
156 Symmetry is what we
102 Fourteen forms with all
their branches and cases see at a glance
Cubic equations Pascal’s triangle

106 The ubiquitous music 162 Chance is bridled and

governed by law
of the spheres
The Fibonacci sequence THE RENAISSANCE Probability

112 The power of doubling

1500–1680 166 The sum of the distance
Wheat on a chessboard equals the altitude
118 The geometry of art Viviani’s triangle theorem
and life
The golden ratio
167 The swing of a
pendulum
124 Like a large diamond
Mersenne primes Huygens’s tautochrone
curve
125 Sailing on a rhumb
Rhumb lines

126 A pair of equal-length

lines
The equals sign and other
symbology

128 Plus of minus times plus

of minus makes minus
Imaginary and complex
numbers

132 The art of tenths

Decimals

138 Transforming
multiplication
into addition
Logarithms
8

168 With calculus I can predict

the future
Calculus
THE 19TH CENTURY
1800–1900
176 The perfection of the
science of numbers 214 Complex numbers are
Binary numbers coordinates on a plane
The complex plane
216 Nature is the most fertile
THE source of mathematical
discoveries
ENLIGHTENMENT Fourier analysis

1680–1800 218 The imp that knows the

positions of every particle
182 To every action there is
in the Universe
an equal and opposite Laplace’s demon
reaction
Newton’s laws of motion 220 What are the chances?
The Poisson distribution
184 Empirical and expected
results are the same 221 An indispensable tool
The law of large numbers in applied mathematics
194 The seven bridges of Bessel functions
186 One of those strange Königsberg
numbers that are Graph theory 222 It will guide the future
creatures of their own course of science
Euler’s number 196 Every even integer is
The mechanical computer
the sum of two primes
192 Random variation makes The Goldbach conjecture
a pattern
Normal distribution 197 The most beautiful
equation
Euler’s identity

198 No theory is perfect

Bayes’ theorem

200 Simply a question

of algebra
The algebraic resolution
of equations

202 Let us gather facts

Buffon’s needle experiment

204 Algebra often gives

more than is asked of her
The fundamental theorem
of algebra
 9

254 A diagrammatic
representation of
reasonings
Venn diagrams

255 The tower will fall

and the world will end
The Tower of Hanoi

256 Size and shape do not

matter, only connections
Topology

260 Lost in that silent,

measured space
The prime number theorem

226 A new kind of function

Elliptic functions
MODERN
228 I have created another
world out of nothing MATHEMATICS 272 A freer logic
Non-Euclidean geometries 1900–PRESENT emancipates us
The logic of mathematics
230 Algebraic structures 266 The veil behind which
have symmetries the future lies hidden 274 The Universe is
Group theory 23 problems for the four-dimensional
20th century Minkowski space
234 Just like a pocket map
Quaternions 268 Statistics is the grammar 276 Rather a dull number
of science Taxicab numbers
236 Powers of natural The birth of modern
numbers are almost statistics 278 A million monkeys
never consecutive
Catalan’s conjecture banging on a million
typewriters
238 The matrix is everywhere
The infinite monkey theorem
Matrices
280 She changed the face
242 An investigation into the of algebra
laws of thought Emmy Noether and
Boolean algebra abstract algebra

248 A shape with just one side 282 Structures are the
The Möbius strip weapons of the
mathematician
250 The music of the primes The Bourbaki group
The Riemann hypothesis
284 A single machine to
252 Some infinities are bigger compute any computable
than others sequence
Transfinite numbers The Turing machine
10

305 Pentagons are just nice

to look at
The Penrose tile

306 Endless variety and

unlimited complication
Fractals

312 Four colours but no more

The four-colour theorem

314 Securing data with a

one-way calculation
Cryptography

318 Jewels strung on an

as-yet invisible thread
Finite simple groups

320 A truly marvellous proof

Proving Fermat’s last
theorem

324 No other recognition

290 Small things are more
numerous than large
is needed 326 DIRECTORY
Proving the Poincaré
things conjecture
Benford’s law

291 A blueprint for the

336 GLOSSARY
digital age
Information theory

292 We are all just six steps

344 INDEX
away from each other
Six degrees of separation

294 A small positive vibration

351 QUOTATIONS
can change the entire
cosmos
The butterfly effect 352 ACKNOWLEDGEMENTS
300 Logically things can
only partly be true
Fuzzy logic

302 A grand unifying theory

of mathematics
The Langlands Program

304 Another roof, another

proof
Social mathematics
 11

FOREWORD
Summarizing all of mathematics in one book is a where the sum in each row, column, and diagonal
daunting and indeed impossible task. Humankind is always the same – are one of the oldest areas of
has been exploring and discovering mathematics recreational mathematics. Starting in the 9th century
for millennia. Practically, we have relied on maths BCE in China, the story then bounces around via
to advance our species, with early arithmetic and Indian texts from 100 CE, Arab scholars in the Middle
geometry providing the foundations for the first cities Ages, Europe during the Renaissance, and finally,
and civilizations. And philosophically, we have used modern Sudoku-style puzzles. Across a mere two
mathematics as an exercise in pure thought to explore pages this book has to cover 3,000 years of history
patterns and logic. ending with geomagic squares in 2001. And even in
As a subject, mathematics is surprisingly hard to this small niche of mathematics, there will be many
pin down with one catch-all definition. “Mathematics” magic square developments that there was simply not
is not simply, as many people think, “stuff to do enough room to include. The whole book should be
with numbers”. That would exclude a huge range of viewed as a curated tour of mathematical highlights.
mathematical topics, including much of the geometry Studying even just a sample of mathematics is a
and topology covered in this book. Of course numbers great reminder of how much humans have achieved.
are still very useful tools to understand even the most But it also highlights where mathematics could do
esoteric areas of mathematics, but the point is that better: things like the glaring omission of women
they are not the most interesting aspect of it. Focusing from the history of mathematics cannot be ignored.
just on numbers misses the wood for the threes. A lot of talent has been squandered over the centuries,
For the record, my own definition of maths as and a lot of credit has not been appropriately given.
“the sort of things that mathematicians enjoy doing”, But I hope that we are now improving the diversity
while delightfully circular, is largely unhelpful. Big of mathematicians, and encouraging all humans to
Ideas: Simply Explained is actually not a bad definition. discover and learn about mathematics.
Mathematics could be seen as the attempt to find the Because going forward, the body of mathematics
simplest explanations for the biggest ideas. It is the will continue to grow. Had this book been written a
endeavour of finding and summarizing patterns. century earlier it would have been much the same up
Some of those patterns involve the practical triangles until about page 280. And then it would have ended.
required to build pyramids and divide land; other No ring theory from Emmy Noether, no computing from
patterns attempt to classify all of the 26 sporadic Alan Turing, and no six degrees of separation from
groups of abstract algebra. These are very different Kevin Bacon. And no doubt that will be true again
problems in terms of both usefulness and complexity, 100 years from now. The edition printed a century from
but both types of pattern have become the obsession now will carry on past page 325: covering patterns
of mathematicians throughout the ages. totally alien to us. And because anyone can do maths,
There is no definitive way to organize all of there is no telling who will discover this new maths,
mathematics, but looking at it chronologically is not a and where or when. To make the biggest advancement
bad way to go. This book uses the historical journey of in mathematics during the 21st century, we need to
humans discovering maths as a way to classify it and include all people. I hope this book helps inspire
wrangle it into a linear progression. Which is a valiant everyone to get involved.
but difficult effort. Our current mathematical body of
knowledge has been built up by a haphazard and
diverse range of people across time and cultures.
So something like the short section on magic
squares covers thousands of years and the span of
the globe. Magic squares – arrangements of numbers Matt Parker
INTRODU
CTION
14 INTRODUCTION

T
he history of mathematics invention. Although it was additional items, or depletion of
reaches back to prehistory, human curiosity and intuition a stock, the basic operations
when early humans found that recognized the underlying of arithmetic.
ways to count and quantify things. principles of mathematics, and As hunter-gatherers turned to
In doing so, they began to identify human ingenuity that later provided trade and farming, and societies
certain patterns and rules in the various means of recording and became more sophisticated,
concepts of numbers, sizes, and notating them, those principles arithmetical operations and a
shapes. They discovered the themselves are not a human numeral system became essential
basic principles of addition and invention. The fact that 2 + 2 = 4 tools in all kinds of transactions.
subtraction – for example, that two is true, independent of human To enable trade, stock-taking, and
things (whether pebbles, berries, or existence; the rules of mathematics, taxes in uncountable goods such as
mammoths) when added to another like the laws of physics, are oil, flour, or plots of land, systems of
two invariably resulted in four universal, eternal, and unchanging. measurement were developed,
things. While such ideas may When mathematicians first showed putting a numerical value on
seem obvious to us today, they that the angles of any triangle in a dimensions such as weight and
were profound insights for their flat plane when added together come length. Calculations also became
time. They also demonstrate that to 180°, a straight line, this was not more complex, developing the
the history of mathematics is above their invention: they had simply concepts of multiplication and
all a story of discovery rather than discovered a fact that had always division from addition and
been (and will always be) true. subtraction – allowing the area of
land to be calculated, for example.
Early applications In the early civilizations, these
The process of mathematical new discoveries in mathematics,
discovery began in prehistoric and specifically the measurement
times, with the development of of objects in space, became the
It is impossible to be ways of counting things people foundation of the field of geometry,
a mathematician needed to quantify. At its simplest, knowledge that could be used in
without being a this was done by cutting tally building and tool-making. In using
poet of the soul. marks in a bone or stick, a these measurements for practical
Sofya Kovalevskaya rudimentary but reliable means purposes, people found that certain
Russian mathematician of recording numbers of things. patterns were emerging, which
In time, words and symbols were could in turn prove useful. A simple
assigned to the numbers and the but accurate builder’s square can
first systems of numerals began be made from a triangle with sides
to evolve, a means of expressing of three, four, and five units. Without
operations such as acquisition of that accurate tool and knowledge,
 INTRODUCTION 15

the roads, canals, ziggurats, and civilizations of Mesopotamia and

pyramids of ancient Mesopotamia Egypt, through Greece, China,
and Egypt could not have been built. India, and the Islamic empire to
As new applications for these Renaissance Europe and into
mathematical discoveries were the modern world. As it evolved,
found – in astronomy, navigation, Geometry is knowledge mathematics was also seen to
engineering, book-keeping, of the eternally existent. comprise several distinct but
taxation, and so on – further Pythagoras interconnected fields of study.
patterns and ideas emerged. Ancient Greek mathematician The first field to emerge, and in
The ancient civilizations each many ways the most fundamental,
established the foundations is the study of numbers and
of mathematics through this quantities, which we now call
interdependent process of arithmetic, from the Greek word
application and discovery, but arithmos (“number”). At its most
also developed a fascination with basic, it is concerned with counting
mathematics for its own sake, cosmos, and even suggested and assigning numerical values to
so-called pure mathematics. From that these patterns had mystical things, but also the operations,
the middle of the first millennium properties. Often mathematics was such as addition, subtraction,
bce, the first pure mathematicians therefore seen as a complementary multiplication, and division, that
began to appear in Greece, and discipline to philosophy – many can be applied to numbers. From
slightly later in India and China, of the greatest mathematicians the simple concept of a system of
building on the legacy of the through the ages have also been numbers comes the study of the
practical pioneers of the subject – philosophers, and vice versa – and properties of numbers, and even
the engineers, astronomers, and the links between the two subjects the study of the very concept itself.
explorers of earlier civilizations. have persisted to the present day. Certain numbers – such as the
Although these early constants , e, or the prime and
mathematicians were not so Arithmetic and algebra irrational numbers – hold a special
concerned with the practical So began the history of mathematics fascination and have become the
applications of their discoveries, as we understand it today – the subject of considerable study.
they did not restrict their studies discoveries, conjectures, and Another major field in
to mathematics alone. In their insights of mathematicians that mathematics is algebra, which
exploration of the properties of form the bulk of this book. As well is the study of structure, the way
numbers, shapes, and processes, as the individual thinkers and their that mathematics is organized,
they discovered universal rules and ideas, it is a story of societies and and therefore has some relevance
patterns that raised metaphysical cultures, a continuously developing in every other field. What marks
questions about the nature of the thread of thought from the ancient algebra from arithmetic is the ❯❯
16 INTRODUCTION

use of symbols, such as letters, and logic were accepted as the

to represent variables (unknown subject’s foundation for some 2,000
numbers). In its basic form, algebra years. In the 19th century, however,
is the study of the underlying rules alternatives to classical Euclidean
of how those symbols are used in geometry were proposed, opening
mathematics – in equations, for In mathematics, the art up new areas of study, including
example. Methods of solving of asking questions is topology, which examines the
equations, even quite complex more valuable than nature and properties not only of
quadratic equations, had been solving problems. objects in space, but of space itself.
discovered as early as the ancient Georg Cantor Since the Classical period,
Babylonians, but it was medieval German mathematician mathematics had been concerned
mathematicians of the Islamic with static situations, or how
Golden Age who pioneered the use things are at any given moment. It
of symbols to simplify the process, failed to offer a means of measuring
giving us the word “algebra”, which or calculating continuous change.
is derived from the Arabic al-jabr. Calculus, developed independently
More recent developments in by Gottfried Leibniz and Isaac
algebra have extended the idea calendars. A particular branch of Newton in the 17th century,
of abstraction into the study of geometry, trigonometry (the study provided an answer to this problem.
algebraic structure, known as of the properties of triangles), The two branches of calculus,
abstract algebra. proved to be especially useful in integral and differential, offered a
these pursuits. Perhaps because of method of analysing such things
Geometry and calculus its very concrete nature, for many as the slope of curves on a graph and
A third major field of mathematics, ancient civilizations, geometry was the area beneath them as a way of
geometry, is concerned with the the cornerstone of mathematics, describing and calculating change.
concept of space, and the and provided a means of problem- The discovery of calculus
relationships of objects in space: solving and proof in other fields. opened up a field of analysis that
the study of the shape, size and This was particularly true of later became particularly relevant
position of figures. It evolved from ancient Greece, where geometry to, for example, the theories of
the very practical business of and mathematics were virtually quantum mechanics and chaos
describing the physical dimensions synonymous. The legacy of great theory in the 20th century.
of things, in engineering and mathematical philosophers such
construction projects, measuring as Pythagoras, Plato, and Aristotle Revisiting logic
and apportioning plots of land, and was consolidated by Euclid, whose The late 19th and early 20th
astronomical observations for principles of mathematics based centuries saw the emergence of
navigation and compiling on a combination of geometry another field of mathematics – the
 INTRODUCTION 17

foundations of mathematics. This “donkey work” of calculation to shoulders of giants”, developing the
revived the link between philosophy provide tables for mathematicians, ideas of previous thinkers, or finding
and mathematics. Just as Euclid astronomers and so on, the actual practical applications for them.
had done in the 3rd century bce, construction of computers required This book presents many of the
scholars including Gottlob Frege new mathematical thinking. It was “big ideas” in mathematics, from
and Bertrand Russell sought to mathematicians, as much as the earliest discoveries to the
discover the logical foundations on engineers, who provided the means present day, explaining them in
which mathematical principles are of building mechanical, and then layperson’s language, where they
based. Their work inspired a electronic computing devices, came from, who discovered them,
re-examination of the nature of which in turn could be used as and what makes them significant.
mathematics itself, how it works, tools in the discovery of new Some may be familiar, others less
and what its limits are. This study mathematical ideas. No doubt, so. With an understanding of
of basic mathematical concepts is new applications for mathematical these ideas, and an insight into
perhaps the most abstract field, a theorems will be found in the future the people and societies in which
sort of meta-mathematics, yet an too – and with numerous problems they were discovered, we can
essential adjunct to every other still unsolved, it seems that there is gain an appreciation of not only
field of modern mathematics. no end to the mathematical the ubiquity and usefulness of
discoveries to be made. mathematics, but also the elegance
New technology, new ideas The story of mathematics is one and beauty that mathematicians
The various fields of mathematics – of exploration of these different find in the subject. ■
arithmetic, algebra, geometry, fields, and the discovery of new
calculus, and foundations – are ones. But it is also the story of the
worthy of study for their own sake, explorers, the mathematicians who
and the popular image of academic set out with a definite aim in mind,
mathematics is that of an almost to find answers to unsolved
incomprehensible abstraction. problems, or to travel into unknown Mathematics, rightly viewed,
But applications for mathematical territory in search of new ideas –
possesses not only truth,
discoveries have usually been and those who simply stumbled
found, and advances in science and upon an idea in the course of their
but supreme beauty.
technology have driven innovations mathematical journey, and were
Bertrand Russell
British philosopher and
in mathematical thinking. inspired to see where it would lead. mathematician
A prime example is the Sometimes the discovery would
symbiotic relationship between come as a game-changing
mathematics and computers. revelation, providing a way into
Originally developed as a unexplored fields; at other times
mechanical means of doing the it was a case of “standing on the
ANCIENT A
CLASSICAL
6000 BCE–500 CE
ND
PERIODS
20 INTRODUCTION

Different quantities The ancient Egyptians Hippasus of One of the most influential
are denoted on describe methods of Metapontum discovers textbooks ever written, Euclid’s
Sumerian clay tablets, working out areas and irrational numbers, Elements, contains mathematical
prefiguring a volumes and record them which cannot be advances including proof of the
numerical system. on the Rhind papyrus. expressed in fractions. infinity of prime numbers.

C. 6000 BCE C. 1650 BCE C. 430 BCE C. 300 BCE

C. 4000 BCE C. 530 BCE C. 387 BCE

The Babylonians introduce a Pythagoras founds a Plato founds his Academy

base-60 numerical system, school where he teaches in Athens – the sign over
in which a small cone his metaphysical beliefs the entrance reads “Let
denotes 1 and a large and mathematical no one ignorant of
cone denotes 60. discoveries, including geometry enter here”.
Pythagoras’s theorem.

A
s early as 40,000 years geometry and algebra to solve They established the study of
ago, humans were making practical problems – such as the principles of arithmetic and
tally marks on wood and building, engineering, and geometry as early as 2000 bce.
bone as a means of counting. They calculating land divisions –
undoubtedly had a rudimentary alongside the arithmetical skills Greek rigour
sense of number and arithmetic, they used to conduct commerce The 6th century bce onwards saw
but the history of mathematics and levy taxes. a rapid rise in the influence of
only properly began with the A similar story emerges in ancient Greece across the eastern
development of numerical systems the slightly later civilization of the Mediterranean. Greek scholars
in early civilizations. The first of ancient Egyptians. Their trade and quickly assimilated the
these emerged in the sixth taxation required a sophisticated mathematical ideas of the
millennium bce, in Mesopotamia, numerical system, and their Babylonians and Egyptians. The
western Asia, home to the world’s building and engineering works Greeks used a numerical system of
earliest agriculture and cities. relied on both a means of base-10 (with ten symbols) derived
Here, the Sumerians elaborated measurement and some knowledge from the Egyptians. Geometry in
on the concept of tally marks, of geometry and algebra. The particular chimed with Greek
using different symbols to denote Egyptians were also able to use culture, which idolized beauty of
different quantities, which the their mathematical skills in form and symmetry. Mathematics
Babylonians then developed into conjunction with observations became a cornerstone of Classical
a sophisticated numerical system of the heavens to calculate and Greek thinking, reflected in its art,
of cuneiform (wedge-shaped) predict astronomical and seasonal architecture, and even philosophy.
characters. From about 4000 bce, cycles and construct calendars for The almost mystical qualities of
the Babylonians used elementary the religious and agricultural year. geometry and numbers inspired
 ANCIENT AND CLASSICAL PERIODS 21

The ancient Chinese develop Liu Hui produces an important

Key advances in a system to represent commentary on The Nine Chapters
geometry are made by negative and positive on the Mathematical Art, compiled
Apollonius of Perga numbers using black and by Chinese scholars from as early
in his Conics. red bamboo rods. as the 10th century bce.

C. 200 BCE C. 150 BCE 263

C. 250 BCE C. 150 BCE C. 250 CE 470

Archimedes Hipparchus of Nicaea Diophantus invents new Zu Chongzhi approximates

approximates pi compiles the first symbols to replace pi to seven decimal
by using trigonometric tables. unknown powers in places, a calculation
polygons. equations, and publishes unsurpassed for the
them in his Arithmetica. next millennium.

Pythagoras and his followers to Plato himself described the five foundation of mathematics for the
establish a cult-like community, Platonic solids (the tetrahedron, next two millennia. With similar
dedicated to studying the cube, octahedron, dodecahedron, rigour, Diophantus pioneered the
mathematical principles they and icosahedron). Other use of symbols to represent
believed were the foundations of philosophers, notably Zeno of Elea, unknown numbers in his equations;
the Universe and everything in it. applied logic to the foundations of this was the first step towards the
Centuries before Pythagoras, mathematics, exposing the symbolic notation of algebra.
the Egyptians had used a triangle problems of infinity and change.
with sides of 3, 4, and 5 units as They even explored the strange A new dawn in the East
a building tool to ensure corners phenomenon of irrational numbers. Greek dominance was eventually
were square. They had come across Plato’s pupil Aristotle, with his eclipsed by the rise of the Roman
this idea by observation, and then methodical analysis of logical Empire. The Romans regarded
applied it as a rule of thumb, forms, identified the difference mathematics as a practical tool
whereas the Pythagoreans set between inductive reasoning (such rather than worthy of study. At the
about rigorously showing the as inferring a rule of thumb from same time, the ancient civilizations
principle, offering a proof that it is observations) and deductive of India and China independently
true for all right-angled triangles. reasoning (using logical steps to developed their own numerical
It is this notion of proof and rigour reach a certain conclusion from systems. Chinese mathematics in
that is the Greeks’ greatest established premises, or axioms). particular flourished between the
contribution to mathematics. From this basis, Euclid laid out 2nd and 5th centuries ce, thanks
Plato’s Academy in Athens the principles of mathematical largely to the work of Liu Hui in
was dedicated to the study of proof from axiomatic truths in his revising and expanding the classic
philosophy and mathematics, and Elements, a treatise that was the texts of Chinese mathematics. ■
NUMERALS
TAKE THEIR
PLACES
POSITIONAL NUMBERS
24 POSITIONAL NUMBERS

IN CONTEXT
A system of numbers is required
KEY CIVILIZATION to record quantifiable information.
Babylonians
FIELD
Arithmetic
BEFORE By placing the same
It is impractical to
40,000 years ago Stone symbols in different
give every number
Age people in Europe and positions, information
its own symbol.
Africa count using tally marks is conveyed efficiently.
on wood or bone.
6000–5000 bce Sumerians
develop early calculation
Only ten
systems to measure land The position
indicates a symbol’s symbols are needed
and to study the night sky.
numerical value. to represent every
4000–3000 bce Babylonians number.
use a small clay cone for 1 and
a large cone for 60, together

T
with a clay ball for 10, as their
he first people known to the ancient Egyptians used separate
base-60 system evolves.
have used an advanced symbols for ones, tens, hundreds,
AFTER numeration system were thousands, and above, and had no
2nd century ce The Chinese the Sumerians of Mesopotamia, an place value system. Representing
use an abacus in their base-10 ancient civilization living between larger numbers could require 50 or
positional number system. the Tigris and Euphrates rivers in more hieroglyphs.
what is present-day Iraq. Sumerian
7th century In India, clay tablets from as early as the Using different bases
Brahmagupta establishes zero 6th millennium bce include symbols The Hindu–Arabic numeration
as a number in its own right denoting different quantities. that is employed today is a base-10
and not just a placeholder. The Sumerians, followed by the (decimal) system. It requires only
Babylonians, needed efficient 10 symbols – nine digits (1, 2,
mathematical tools in order to 3, 4, 5, 6, 7, 8, 9) and a zero as a
administer their empires. placeholder. As in the Babylonian
What distinguished the system, the position of a digit
Babylonians from neighbours indicates its value, and the smallest
such as Egypt was their use of a value digit is always to the right.
positional (place value) number In a base-10 system, a two-digit
It is given to us to calculate, system. In such systems, the value number, such as 22, indicates
to weigh, to measure, of a number is indicated both by (2  101) + 2; the value of the 2
to observe; this is its symbol and its position. Today, on the left is ten times that of the
for instance, in the decimal system, 2 on the right. Placing digits after
natural philosophy.
the position of a digit in a number the number 22 will create hundreds,
Voltaire indicates whether its value is in thousands, and larger powers of
French philosopher
ones (less than 10), tens, hundreds, 10. A symbol after a whole number
or more. Such systems make (the standard notation now is a
calculation more efficient because a decimal point) can also separate
small set of symbols can represent a it from its fractional parts, each
huge range of values. By contrast, representing a tenth of the place
 ANCIENT AND CLASSICAL PERIODS 25
See also: The Rhind papyrus 32–33 ■ The abacus 58–59 ■ Negative numbers
76–79 ■ Zero 88–91 ■ The Fibonacci sequence 106–11 ■ Decimals 132–37
Cuneiform
In the late 19th century,
academics deciphered the
value of the preceding figure. to represent symbols for 1 to 9. For
“cuneiform” (wedge-shaped)
The Babylonians worked with 10, a different symbol was used, markings on clay tablets
a more complex sexagesimal placed to the left of the one symbol, recovered from Babylonian
(base-60) number system that and repeated two to five times in sites in and around Iraq. Such
was probably inherited from the numbers up to 59. At 60 (60  1), marks, denoting letters and
earlier Sumerians and is still the original symbol for one was words as well as an advanced
used across the world today for reused but placed further to the left number system, were etched
measuring time, degrees in a circle than the symbol for 1. Because it in wet clay with either end of
(360° = 6  60), and geographic was a base-60 system, two such a stylus. Like the Egyptians,
coordinates. Why they used 60 as symbols signified 61, while three the Babylonians needed
a number base is still not known such symbols indicated 3,661, that scribes to administer their
for sure. It may have been chosen is, 60  60 (602) + 60 + 1. complex society, and many
because it can be divided by many The base-60 system had of the tablets bearing
other numbers – 1, 2, 3, 4, 5, 6, 10, obvious drawbacks. It necessarily mathematical records are
thought to be from training
12, 15, 20, and 30. The Babylonians requires many more symbols than
schools for scribes.
also based their calendar year on a base-10 system. For centuries, the
A great deal has now been
the solar year (365.24 days); the sexagesimal system also had no ❯❯ discovered about Babylonian
number of days in a year was mathematics, which extended
360 (6  60) with additional days to multiplication, division,
The Babylonian sun-god Shamash
for festivals. awards a rod and a coiled rope, ancient geometry, fractions, square
In the Babylonian sexagesimal measuring devices, to newly trained roots, cube roots, equations,
system, a single symbol was used surveyors, on a clay tablet dating from and other forms, because –
alone and repeated up to nine times around 1000 bce. unlike Egyptian papyrus
scrolls – the clay tablets
have survived well. Several
thousand, mostly dating from
between 1800 and 1600 bce,
are housed in museums
around the world.

Cuneiform, a word derived

from the Latin cuneus (“wedge”)
to describe the shape of the
symbols, was inscribed into
wet clay, stone, or metal.
26 POSITIONAL NUMBERS
vertical rods indicated 405 (4 
1 11 21 31 41 51 100, or 102) + 5  1 – the absence
of horizontal rods meant there were
2 12 22 32 42 52 no tens in the number. Calculations
were carried out by manipulating
3 13 23 33 43 53 the rods on a counting board.
Positive and negative numbers
4 14 24 34 44 54 were represented by red and black
rods respectively or different
5 15 25 35 45 55 cross-sections (triangular and
rectangular). Rod numerals are still
6 16 26 36 46 56 used occasionally in China, just as
Roman numerals are sometimes
7 17 27 37 47 57 used in Western society.
The Chinese place value system
8 18 28 38 48 58 is reflected in the Chinese abacus
(suanpan). Dating back to at least
200 bce, it is one of the oldest bead-
9 19 29 39 49 59
counting devices, although the
Romans used something similar.
10 20 30 40 50 60 The Chinese version, which is still
used today, has a central bar and a
The Babylonian base-60 number system was built from two varying number of vertical wires to
symbols – the single unit symbol, used alone and combined for separate ones from tens, hundreds,
numbers 1 to 9, and the 10 symbol, repeated for 20, 30, 40, and 50. or more. In each column, there are
two beads above the bar worth five
place value holders, and nothing expressed as quatre-vingt (4  20); each and five beads below the bar
to separate whole numbers from Welsh and Irish also express some worth one each.
fractional parts. By around 300 bce, numbers as multiples of 20, while The Japanese adopted the
however, the Babylonians used in English a score is 20. In the Bible, Chinese abacus in the 14th century
two wedges to indicate no value, for instance, Psalm 90 talks of a and developed their own abacus,
much as we use a placeholder zero human lifespan being “threescore the soroban, which has one bead
today; this was possibly the earliest years and ten” or as great as worth five above the central bar
use of zero. “fourscore years”.
From around 500 bce until the
Other counting systems 16th century when Hindu–Arabic
In Mesoamerica, on the other side numbers were officially adopted
of the world, the Mayan civilization in China, the Chinese used rod
developed its own advanced numerals to represent numbers. The Babylonian and Assyrian
numeration system in the 1st This was the first decimal place civilizations have perished…
millennium bce – apparently in value system. By alternating yet Babylonian mathematics
complete isolation. Theirs was a quantities of vertical rods with is still interesting, and the
base-20 (vigesimal) number horizontal rods, this system could
Babylonian scale of 60 is still
system, which probably evolved indicate ones, tens, hundreds,
from a simple counting method thousands, and more powers of
used in astronomy.
using fingers and toes. In fact, 10, much as the decimal system
G. H. Hardy
British mathematician
base-20 number systems were does today. For example, 45 was
used across the world, in Europe, written with four horizontal bars
Africa, and Asia. Language often representing 4  101 (40) and five
contains remnants of this system. vertical bars for 5  1 (5). However,
For example, in French, 80 is four vertical rods followed by five
 ANCIENT AND CLASSICAL PERIODS 27
Ebisu, the Japanese god of fishermen
and one of the seven gods of fortune,
uses a soroban to calculate his profits
in The Red Snapper’s Dream by
Utagawa Toyohiro.
The fact that we work in 10s as
opposed to any other number to be written efficiently, through
is purely a consequence of our the use of place value. The system
anatomy. We use our ten was adopted and refined by Arab
fingers to count. mathematicians in the 9th century.
Marcus du Sautoy They introduced the decimal point,
British mathematician so that the system could also
express fractions of whole numbers.
Three centuries later, Leonardo
of Pisa (Fibonacci) popularized the
use of Hindu–Arabic numerals in
Europe through his book Liber
Abaci (1202). Yet the debate about
and four beads each worth one whether to use the new system
below the bar in each column. Japan rather than Roman numerals and
still uses the soroban today: there traditional counting methods 2, also expressed as 21, 22, 23 and
are even contests in which young lasted for several hundred years, upwards. In binary, the number
people demonstrate their ability before its adoption paved the way 111 means 1  22 + 1  21 + 1  20,
to perform soroban calculations for modern mathematical advances. that is 4 + 2 + 1, or 7 in our decimal
mentally, a skill known as anzan. With the advent of electronic number system.
computers, other number bases In binary, as in all modern
Modern numeration became important – particularly number systems whatever their
The Hindu–Arabic decimal system binary, a number system with base, the principles of place value
used throughout the world today base 2. Unlike the base-10 system are always the same. Place value –
has its origins in India. In the 1st with its 10 symbols, binary has the Babylonian legacy – remains
to 4th centuries ce the use of nine just two: 1 and 0. It is a positional a powerful, easily understood,
symbols along with zero was system but instead of multiplying and efficient way to represent
developed to allow any number by 10, each column is multiplied by large numbers. ■

Mayan numeral system bars they could generate

numerals up to 19. Numbers
The Mayans, who lived in Central larger than 19 were written
America from around 2000 bce, vertically, with the lowest
used a base-20 (vigesimal) number numbers at the bottom, and
system from around 1000 bce to there is evidence of Mayan
perform astronomical and calendar calculations up to hundreds of
calculations. Like the Babylonians, millions. An inscription from
they used a calendar of 360 days 36 bce shows that they used a
plus festivals, to make 365.24 days shell-shaped symbol to denote
based on the solar year; their zero, which was widely used
calendars helped them work out by the 4th century.
the growing cycles of crops. The Mayans’ number system
The Dresden Codex, the oldest The Mayan system employed was in use in Central America
surviving Mayan book, dating from symbols: a dot representing one until the Spanish conquests in
the 13th or 14th century, illustrates and a bar representing five. By the 16th century. Its influence,
Mayan number symbols and glyphs. using combinations of dots over however, never spread further.
28
IN CONTEXT

THE SQUARE
KEY CIVILIZATIONS
Egyptians (c. 2000 bce),
Babylonians (c. 1600 bce)

AS THE
FIELD
Algebra
BEFORE
c. 2000 bce The Berlin papyrus

HIGHEST
records a quadratic equation
solved in ancient Egypt.
AFTER
7th century ce The Indian

POWER
mathematician Brahmagupta
solves quadratic equations
using only positive integers.
10th century ce Egyptian
scholar Abu Kamil Shuja ibn

QUADRATIC EQUATIONS Aslam uses negative and

irrational numbers to solve
quadratic equations.
1545 Italian mathematician
Gerolamo Cardano publishes
his Ars Magna, setting out the
rules of algebra.

Q
uadratic equations are
those involving unknown
numbers to the power of
2 but not to a higher power; they
contain x2 but not x3, x4, and so on.
One of the main rudiments of
mathematics is the ability to use
equations to work out solutions to
real-world problems. Where those
problems involve areas or paths of
curves such as parabolas, quadratic
equations become very useful, and
describe physical phenomena, such
as the flight of a ball or a rocket.

Ancient roots
The history of quadratic equations
extends across the world. It is likely
that these equations first arose
 ANCIENT AND CLASSICAL PERIODS 29
See also: Irrational numbers 44–45 ■ Negative numbers 76–79 ■ Diophantine equations 80–81 ■ Zero 88–91 ■ Algebra
92–99 ■ The binomial theorem 100–01 ■ Cubic equations 102–05 ■ Imaginary and complex numbers 128–31

Quadratic equations contain the power of 2, so are

used when calculating with two dimensions.

The number of dimensions is equal to the maximum

number of real solutions an equation has.

There is a maximum of two real solutions for a quadratic

equation, three for a cubic equation, and so on.

If a quadratic equation, or any equation, is set equal to zero

(e.g. x2 + 3x + 2 = 0), the solutions are called roots.

The Berlin papyrus was copied and

published by German Egyptologist
Hans Schack-Schackenburg in 1900. It
In a quadratic equation, these two roots are the points where a contains two mathematical problems,
quadratic curve crosses the x-axis on a graph. one of which is a quadratic equation.

3 is used because this length is

from the need to subdivide land the quadratic equation (3/4 y)2 + 3/4 of the side of the other small

for inheritance purposes, or to y2 = 100 to find the length of a square. Two squares created
solve problems involving addition side on each square. using these false position numbers
and multiplication. The Egyptians used a method would have areas of 16 and 9
One of the oldest surviving called “false position” to determine respectively, which when added
examples of a quadratic equation the solution. In this method, the together give a total area of 25.
comes from the ancient Egyptian mathematician selects a convenient This is only 1/4 of 100, so the areas
text known as the Berlin papyrus number that is usually easy to must be quadrupled to match the
(c. 2000 bce). The problem contains calculate, then works out what the Berlin papyrus equation. The
the following information: the area solution to the equation would be lengths therefore must be doubled
of a square of 100 cubits is equal using that number. The result from the false positions of 4 and 3
to that of two smaller squares. shows how to adjust the number to reach the solutions: 8 and 6.
The side of one of the smaller to give the correct solution the Other early records of quadratic
squares is equal to one half plus a equation. For example, in the Berlin equations are found in Babylonian
quarter of the side of the other. In papyrus problem, the simplest clay tablets, where the diagonal of
modern notation, this translates length to use for the larger of the a square is given to five decimal
into two simultaneous equations: two small squares is 4, because places. The Babylonian tablet YBC
x2 + y2 = 100 and x = (1/2 + 1/4)y the problem deals with quarters. 7289 (c. 1800–1600 bce) shows a
= 3/4 y. These can be simplified to For the side of the smallest square, method of working out the ❯❯
30 QUADRATIC EQUATIONS
The quadratic formula is a way to solve quadratic equations. By modern multiplied together make a positive
convention, quadratic equations include a number, a, multiplied by x2; number. While M2  M2 = 2, it is
a number, b, multiplied by x; and a number, c, on its own. The illustration also true that -M2 -M2 = 2.
below shows how the formula uses a, b, and c to find the value of x.
Quadratic equations often equal 0, because this makes them easy to work
In 1545, Italian scholar Gerolamo
out on a graph; the x solutions are the points where the curve crosses the x axis. Cardano published his Ars Magna
(The Great Art, or the Rules of
Algebra) in which he explored the
Number that Number on problem: “What pair of numbers
multiplies
x its own have a sum of ten and product of
QUADRATIC EQUATION 40?” He found that the problem led
Number that ax2 + bx + c = 0 to a quadratic equation which,
multiplies 2
x
when he completed the square,
QUADRATIC FORMULA gave √(-15). No numbers available
to mathematicians at the time gave
x = -b ±b2 - 4ac a negative number when multiplied
KEY by themselves, but Cardano
Input of a into the formula Plus or
2a suggested suspending belief and
minus working with the square root of
Input of b into the formula
Input of c into the formula negative 15 to find the equation’s
two solutions. Numbers such as
√(-15) would later be known as
“imaginary” numbers.
quadratic equation x2 = 2 by is one of two correct solutions
drawing rectangles and trimming to the problem; -13 is the other. If Structure of equations
them down into squares. In the 7th x is -13, x2 = 169 and 10x = -130. Modern quadratic equations usually
century ce, Indian mathematician Adding a negative number gives look like ax2 + bx + c = 0. The
Brahmagupta wrote a formula for the same result as subtracting its letters a, b, and c represent known
solving quadratic equations that equivalent positive number, so numbers, while x represents the
could be applied to equations in the 169 + -130 = 169 - 130 = 39. unknown number. Equations
form ax2 + bx = c. Mathematicians In the 10th century, Egyptian contain variables (symbols for
at the time did not use letters or scholar Abu Kamil Shuja ibn Aslam numbers that are unknown),
symbols, so he wrote his solution made use of negative numbers and coefficients, constants (those that
in words, but it was similar to the algebraic irrational numbers (such do not multiply variables), and
modern formula shown above. as the square root of 2) as both operators (symbols such as the
In the 8th century, Persian solutions and coefficients (numbers plus and equals sign). Terms are
mathematician al-Khwarizmi multiplying an unknown quantity). the parts separated by operators;
employed a geometric solution By the 16th century, most
for quadratic equations known as mathematicians accepted negative
completing the square. Until the solutions and were comfortable with
10th century, geometric methods surds (irrational roots – those that
were were often used, as quadratic cannot be expressed exactly as a
equations were used to solve real- decimal). They had also started
world problems involving land rather using numbers and symbols, rather Politics is for the present,
than abstract algebraic challenges. than writing equations in words. but an equation
Mathematicians now utilized the is for eternity.
Negative solutions plus or minus symbol, ±, in solving Albert Einstein
Indian, Persian, and Arab scholars quadratic equations. With the
thus far had used only positive equation x2 = 2, the solution is not
numbers. When solving the just x = M2 but x = ±M2. The plus
equation x2 + 10x = 39, they gave or minus symbol is included
the solution as 3. However, this because two negative numbers
 ANCIENT AND CLASSICAL PERIODS 31
y A graph of the
13 quadratic function Practical applications
y = ax2 + bx + c
(-5, 12) 12 (2, 12) creates a U-shaped Although they were initially
curve called a used for working out geometric
11 parabola. This graph problems, today quadratic
plots the points (in equations are important in
10 black) of the quadratic many aspects of mathematics,
function where a = 1, science, and technology.
9 b = 3, and c = 2. Projectile flight, for example,
This expresses the
8 quadratic equation can be modelled with quadratic
x2 + 3x + 2 = 0. The equations. An object thrown
7 up into the air will fall back
solutions for x are
where y = 0 and the down again as a result of
(-4, 6) 6 (1, 6) gravity. The quadratic function
curve crosses the x
5 axis. These are -2 can predict projectile motion –
and -1. the height of the object over
4 time. Quadratic equations are
used to model the relationship
3 between time, speed, and
distance, and in calculations
(-3, 2) 2 (0, 2) with parabolic objects such as
lenses. They can also be used
1
to forecast profits and loss in
(-2, 0) (-1, 0) x the world of business. Profit is
-6 -5 -4 -3 -2 -1 0 1 2 3 based on total revenue minus
production cost; companies
create a quadratic equation
they can be a number or variable, or Parabolas prove useful in the real known as the profit function
a combination of both. The modern world because of their reflective. with these variables to work
quadratic equation has four terms: properties. Satellite dishes are out the optimal sale prices to
ax2, bx, c, and 0. parabolic for this reason. Signals maximize profits.
received by the dish will reflect off
Parabolas the parabola and be directed to
A function is a group of terms one single point – the receiver. ■
that defines a relationship between
variables (often x and y). The
Parabolic Rays of light
quadratic function is generally mirror
written as y = ax2 + bx + c, which,
on a graph, produces a curve called
a parabola (see above). When real
(not imaginary) solutions to ax2 +
bx + c = 0 exist, they will be the A
roots – the points where the parabola
crosses the x axis. Not all parabolas
cut the x axis in two places. If the Line of
parabola touches the x axis only symmetry
once, this means that there are Quadratic equations are used
coincident roots (the solutions are by military specialists to model
Parabolic objects have special the trajectory of projectiles fired by
equal to each other). The simplest reflective properties. With a parabolic artillery – such as this MIM-104
equation of this form is y = x2. If the mirror, any ray of light parallel to its Patriot surface-to-air missile,
parabola does not touch or cross line of symmetry will reflect off the commonly used by the US Army.
the x axis, there are no real roots. surface to the same fixed point (A).
32

THE ACCURATE
RECKONING FOR
INQUIRING INTO
ALL THINGS
THE RHIND PAPYRUS

T
he Rhind papyrus in the earlier Moscow papyrus, illustrated
IN CONTEXT British Museum in London techniques for working out areas,
provides an intriguing proportions, and volumes.
KEY CIVILIZATION
account of mathematics in ancient
Ancient Egyptians
Egypt. Named after Scottish Representing concepts
(c. 1650 bce)
antiquarian Alexander Henry The Egyptian number system was
FIELD Rhind, who purchased the papyrus the first decimal system. It used
Arithmetic in Egypt in 1858, it was copied strokes for single digits and a
from earlier documents by a scribe, different symbol for each power
BEFORE Ahmose, more than 3,500 years of 10. The symbols were then
c. 2480 bce Stone carvings ago. It measures 32 cm (121 ⁄2 in) repeated to create other numbers.
record flood levels on the River by 200 cm (781 ⁄2 in) and includes A fraction was shown as a number
Nile, measured in cubits – 84 problems concerned with with a dot above it. The Egyptian
about 52 cm (201 ⁄2 in) – and arithmetic, algebra, geometry, concept of a fraction was closest to
palms – about 7.5 cm (3 in). and measurement. The problems, a unit fraction – that is 1 ⁄n, where n
recorded in this and other ancient is a whole number. When a fraction
c. 1800 bce The Moscow Egyptian artefacts such as the was doubled, it had to be rewritten
papyrus provides solutions as one unit fraction added to
to 25 mathematical problems, another unit fraction; for example,
including the calculation of the 2 ⁄ 3 in modern notation would be 1 ⁄2
surface area of a hemisphere + 1 ⁄6 in Egyptian notation (not 1 ⁄3 +
and the volume of a pyramid. 1 ⁄ 3 because the Egyptians did not

allow repeats of the same fraction).

AFTER
The 84 problems in the Rhind
c. 1300 bce The Berlin papyrus
papyrus illustrate the mathematical
is produced. It shows that the
methods in common use in ancient
ancient Egyptians used Egypt. Problem 24, for example, asks
quadratic equations. what quantity, if added to its seventh
6th century bce The Greek part, becomes 19. This translates as
The Eye of Horus, an Egyptian god, x + x ⁄ 7 = 19. The approach applied
scientist Thales travels to was a symbol of power and protection.
Egypt and studies its Parts of it were also used to denote to problem 24 is known as “false
mathematical theories. fractions whose denominators were position”. This technique – used well
powers of 2. The eyeball, for example, into the Middle Ages – is based on
represents 1 ⁄4, while the eyebrow is 1 ⁄8. trial and improvement, choosing the
 ANCIENT AND CLASSICAL PERIODS 33
See also: Positional numbers 22–27 Pythagoras 36–43 Calculating pi 60–65
■ Algebra 92–99 ■ Decimals 132–37
■ ■
Instruction books
The Rhind and Moscow
papyri are the most complete
mathematical documents to
10 survive from the height of the
Heel bone ancient Egyptian civilization.
1 2 3 4 5 6 7 8 9 They were painstakingly
copied by scribes well versed
100 in arithmetic, geometry, and
Coiled rope mensuration (the study of
measurements), and are likely
to have been used for training
of other scribes. Although they
captured probably the most
advanced mathematical
knowledge of the time, they
1,000 10,000 100,000 1,000,000 were not seen as works of
Water lily Bent finger Tadpole/frog God Heh scholarship. Instead, they
Ancient Egyptians used vertical lines to denote the numbers were instruction manuals
1 to 9. Powers of ten, particularly those inscribed on stone, were for use in trade, accounting,
depicted as hieroglyphs – picture symbols. construction, and other
activities that involved
measurement and calculation.
simplest, or “false”, value for a whose side length is 8 ⁄9 of the Egyptian engineers, for
variable and adjusting the value diameter, and then multiplies this example, used mathematics
using a scaling factor (the required by the height. The figure of 8 ⁄9 is in the building of pyramids.
quantity divided by the result). used as an approximation for the The Rhind papyrus includes
In the workings for problem 24, proportion of the area of a square a calculation for the slope of a
one-seventh is easiest to find for that would be taken up by a circle pyramid, using the seked –
the number 7, so 7 is used first as a if it were drawn within the square. a measure for the horizontal
“false” value for the variable. The This method is used in problem 50 distance travelled by a slope
result of the calculation – 7 plus 7⁄ 7 to find the area of a circle: subtract for each drop of 1 cubit. The
(or 1) – is 8, not 19, so a scaling 1 ⁄9 from the diameter of the circle, steeper the side of a pyramid,
the fewer the sekeds.
factor is needed. To find how far and find the area of the square with
the guess of 7 is from the required the resulting side length.
quantity, 19 is divided by 8 (the
“false” answer). This produces a Level of accuracy
result of 2 + 1 ⁄4 + 1 ⁄8 (not 23 ⁄8, as Since the Ancient Greeks, the
Egyptian multiplication was based area of a circle has been found by
on doubling and halving fractions), multiplying the square of its radius
which is the scaling factor that (r2) with the number pi (), written
should be applied. So, 7 (the original as r2. The ancient Egyptians had
“false” value) is multiplied by 2 + 1 ⁄4 no concept of pi, but the calculations
+ 1 ⁄8 (the scaling factor) to give the in the Rhind papyrus show that they
quantity 16 + 1 ⁄2 + 1 ⁄8 (or 165 ⁄8). were close to its value. Their circle
Many problems in the papyrus area calculation – with the circle
deal with working out shares of diameter as twice the radius
commodities or land. Problem 41 (2r) – can be expressed as (8 ⁄9  2r)2, The Rhind papyrus scribe used
asks for the volume of a cylindrical which, simplified, is 256 ⁄81 r2, giving the hieratic system of writing
numerals. This cursive style was
store with a diameter of 9 cubits an equivalent for pi of 256 ⁄81. As a more compact and practical than
and a height of 10 cubits. The decimal, this is about 0.6 per cent drawing complex hieroglyphs.
method finds the area of a square greater than the true value of pi. ■
34

THE SUM IS THE

SAME IN EVERY
DIRECTION
MAGIC SQUARES

IN CONTEXT A magic square is a square grid – three-by-three or higher – in

KEY CIVILIZATION which distinct integers have been placed in each cell.
Ancient Chinese
FIELD
Number theory
BEFORE In each row, column, and
This sum is the
diagonal, the sum of the
9th century bce The Chinese magic total.
numbers will be the same.
I Ching (Book of Changes) lays
out trigrams and hexagrams of
numbers for use in divination.

T
here are thousands of ways reference, in the Chinese legend
AFTER
in which to arrange the of Lo Shu (Scroll of the river Lo),
1782 Leonhard Euler writes numbers 1 to 9 in a three- dates from 650 bce. In the legend,
about Latin squares in his by-three grid. Only eight of these a turtle appears to the great King
Recherches sur une nouvelle produce a magic square, where the Yu as he faces a devastating flood.
espèce de carrés magiques sum of the numbers in each row, The markings on the turtle’s back
(Investigations on a new type column, and diagonal – the magic form a magic square, with numbers
of magic square). total – is the same. The sum of from 1 to 9 represented by circular
1979 The first Sudoku-style the numbers 1 to 9 is 45, as is the dots. Because of this legend, the
sum of all three rows or columns. arrangement of odd and even
puzzle is published by Dell
The magic total, therefore, is 1 ⁄3 numbers (even numbers are always
Magazines in New York.
of 45, or 15. In fact, there is really in the corners of the square) were
2001 British electronics just one combination of numbers believed to have magical properties
engineer Lee Sallows in a magic square. The other seven and was used as a good luck
invents magic squares called are rotations of this combination. talisman through the ages.
“geomagic squares”, which As ideas from China spread
contain geometric shapes Ancient origins along trade routes such as the
rather than numbers. Magic squares are probably the Silk Road, other cultures became
earliest example of “recreational interested in magic squares.
mathematics”. Their exact origin Magic squares are discussed in
is unknown, but the first known Indian texts dating from 100 ce,
 ANCIENT AND CLASSICAL PERIODS 35
See also: Irrational numbers 44–45 ■ Eratosthenes’ sieve 66–67 ■ Negative numbers 76–79 ■ The Fibonacci sequence
106–11 ■ The golden ratio 118–23 ■ Mersenne primes 124 ■ Pascal’s triangle 156–61

An order four magic square

appears beneath the bell in Melencolia I 4 9 2 15
by the German artist Albrecht Dürer
and wittily includes the engraving’s
date of 1514.
3 5 7 15

Magic squares have been an

enduring source of fascination for 8 1 6 15
mathematicians. The 15th-century
Italian mathematician Luca Pacioli,
author of De viribus quantitatis (On 15 15 15 15 15
the Power of Numbers), collected The Lo Shu magic square has
magic squares. In 18th-century a magic total of 15.
Switzerland, Leonhard Euler also
became interested in them, and
devised a form that he named Latin 23 20 21 72
squares. The rows and columns in
a Latin square contain figures or
22 24 26 72
symbols that appear only once in
each row and column.
One derivation of the Latin 27 20 72
25
and Brihat-Samhita (c. 550 ce), a square – Sudoku – has become a
book of divination, includes the first popular puzzle. Devised in the US
recorded magic square in India, in the 1970s (where it was called 72 72 72 72 72
used to measure out quantities Number Place), Sudoku took off in Here, 19 is added to each of the
of perfume. Arab scholars, who Japan in the 1980s, acquiring its numbers in the Lo Shu square;
created a vital link between the now-familiar name, which means the magic total is 72.
learning of ancient civilizations “single digits”. A Sudoku puzzle is a
and the European Renaissance, nine-by-nine Latin square with the
introduced magic squares to added restriction that subdivisions 8 18 4 30
Europe in the 14th century. of the square must also contain all
nine numbers. ■
Different sized squares 6 10 14 30
The number of rows and columns
in a magic square is called its order.
For example, a three-by-three magic 16 2 12 30
square is said to have an order of
three. An order two magic square
The most magically magical 30 30 30 30 30
does not exist because it would
only work if all the numbers were
of any magic square ever Here, all the numbers in the Lo
identical. As the orders increase, so
made by a magician. Shu square have been doubled;
do the quantities of magic squares. Benjamin Franklin the magic total is 30.
Talking about a magic square that
Order four produces 880 magic he discovered
squares – with a magic total of 34. Once you have one magic square
you can add the same quantity to every
There are hundreds of millions of number in the square and still end up
order five magic squares, while the with a magic square. Similarly, if you
quantity of order six magic squares multiply all the numbers by the same
has not yet been calculated. quantity you still have a magic square.
NUMBER
IS THE CAUSE OF
GODS AND
DAEMONS
PYTHAGORAS
38 PYTHAGORAS

T
he 6th-century bce Greek
IN CONTEXT philosopher Pythagoras
is also antiquity’s most
KEY FIGURE
famous mathematician. Whether
Pythagoras
or not he was responsible for all the
(c. 570 bce–495 bce) many achievements attributed to
FIELD him in maths, science, astronomy,
Applied geometry music, and medicine, there is no
doubt that he founded an exclusive
BEFORE community that lived for the
c. 1800 bce The columns of pursuit of mathematics and
cuneiform numbers on the philosophy, and regarded numbers
Plimpton 322 clay tablet from as the sacred building blocks of
Babylon include some numbers the Universe.
related to Pythagorean triples.
Angles and symmetry
6th century bce Greek The Pythagoreans were masters of
philosopher Thales of Miletus geometry and knew that the sum of
proposes a non-mythological the three angles of a triangle (180°)
explanation of the Universe – is equal to the sum of two right
pioneering the idea that nature angles (90° + 90°), a fact which two
can be interpreted by reason. Thales of Miletus, one of the Seven
centuries later was described by Sages of ancient Greece, possibly
Euclid as the triangle postulate. inspired the younger Pythagoras with
AFTER
Pythagoras’s followers were also his geometrical and scientific ideas.
c. 380 bce In the tenth book of
aware of some of the regular They may have met in Egypt.
his Republic, Plato espouses polyhedra; these are the perfectly
Pythagoras’s theory of the symmetrical three-dimensional
transmigration of souls. shapes (such as the cube) that were Widely known as Pythagoras’s
c. 300 bce Euclid produces a later known as the Platonic solids. theorem, it states that a2 + b2 = c2,
formula to find sets of primitive Pythagoras himself is primarily where c is the longest side of the
Pythagorean triples. associated with the formula that triangle (the hypotenuse), and a
describes the relationship between and b represent the other two,
the sides of a right-angled triangle. shorter sides that are adjacent

Pythagorean triples of Pythagoras’s schools were

destroyed in a 6th-century bce
The sets of three integers that political purge, Pythagoreans
solve the equation a2 + b2 = c2 emigrated to other parts of
25 are known as Pythagorean triples, southern Italy, spreading their
although their existence was knowledge of triples across the
5
9 known long before Pythagoras. ancient world. Two centuries
3 4 Around 1800 bce, the Babylonians later, Euclid developed a formula
recorded sets of Pythagorean to generate triples: a = m2 - n2,
numbers on the Plimpton 322 clay b = 2mn, c = m2 + n2. With
16 tablet; these show that triples certain exceptions, m and n
become more spread out as the can be any two integers, such
number line progresses. The as 7 and 4, which produce the
The smallest, or most primitive, of Pythagoreans developed methods triple 33, 56, 65 (332 + 562 =
the Pythagorean triples is a triangle for finding sets of triples, and also 652). The formula dramatically
with side lengths 3, 4, and 5. As this proved that there are an infinite sped up the process of finding
graphic shows, 9 plus 16 equals 25. number of such sets. After many new Pythagorean triples.
 ANCIENT AND CLASSICAL PERIODS 39
See also: Irrational numbers 44–45 ■ The Platonic solids 48–49 ■ Syllogistic logic 50–51 ■ Calculating pi 60–65
■ Trigonometry 70–75 ■ The golden ratio 118–23 ■ Projective geometry 154–55

The graphic below demonstrates why the Pythagorean equation (a²+ b²= c²)
works. Within a large square there are four right-angled triangles of equal size
(sides labelled a, b, and c). They are arranged so that a tilted square is formed
in the middle, by the hypotenuses (c sides) of the four triangles.
a b
c a c
b c

c c c
c c b b
a
b a c a
The large square, with area ,
A The smaller, tilted square Each triangle has an area of The total area of the tilted
has a side length of ( + ).
a b inside the large square ab/2 (the base multiplied by
a square plus the triangles
Its area is therefore equal to has an area of 2.c the height and divided by
b is equal to the area of the
( + )2. = ( + ) ( + )
a b A a b a b 2). The total area of all four large square ( ).
A
triangles is 4 ab/2 = 2 .
ab A c= 2+2 ab
is equal to ( + ) ( + ):
A a b a b ( + )( + )=
a b a b c + 2ab 2

Expand the brackets (multiply each term in the first bracket 2 2

a + b + 2ab = c + 2ab 2
by each term in the second bracket). Then add it all together:
Subtract 2ab from each side: 2
a +b =c 2 2

to the right angle. For example, right-angled triangle centuries have reached India. The Egyptians
a right-angled triangle with two before Pythagoras’s birth. However, knew that a triangle with sides of
shorter sides of lengths 3cm and Pythagoras is believed to have 3, 4, and 5 (the first Pythagorean
4cm will have a hypotenuse of been the first to prove the truth triple) would produce a right angle,
length 5cm. The length of this of the formula that states this so their surveyors used ropes of
hypotenuse is found because relationship, and its validity for these lengths to construct perfect
32 + 42 = 52 (9 + 16 = 25). Such sets all right-angled triangles, which is right angles for their building
of whole-number solutions to the why the theorem takes his name. projects. Observing this method
equation a2 + b2 = c2 are known first-hand may have encouraged ❯❯
as Pythagorean triples. Multiplying Journeys of discovery
the triple 3, 4, and 5 by 2 produces Pythagoras was well-travelled, and
another Pythagorean triple: 6, 8, the ideas he absorbed from other
and 10 (36 + 64 = 100). The set countries undoubtedly fuelled his
3, 4, 5 is called a “primitive” mathematical inspiration. Hailing
Pythagorean triple because its from Samos, which was not far from
components share no common Miletus in western Anatolia Reason is
divisor larger than 1. The set (present-day Turkey), he may have immortal, all else
6, 8, 10 is not primitive as its studied at the school of Thales of is mortal.
components share the common Miletus under the philosopher Pythagoras
divisor 2. Anaximander. He embarked on his
There is good evidence that the travels at the age of 20, and spent
Babylonians and the Chinese were many years away. He is thought
well aware of the mathematical to have visited Phoenicia, Persia,
relationship between sides of a Babylon, and Egypt, and may also
40 PYTHAGORAS

Once the
Proving every Instead,
theorem has been
instance of a conjecture mathematicians try to
proved, the truth of
(an unproven theorem) prove the underlying
every instance
would take forever. theorem.
follows.

Pythagoras’s theorem is a clear

example of this process, as it proves that the
sides of every right-angled triangle follow
the rule a2 + b2 = c2.

Pythagoras to study and prove the formed a significant part of its be written as the sum of three
underlying mathematical theorem. 600 members. When they joined, triangular numbers: 1 + 3 + 15 =
In Egypt, Pythagoras may also members were obliged to give all 19. Fermat could not prove this
have met Thales of Miletus, a keen their possessions and wealth to conjecture; it was only in 1813 that
geometrician, who calculated the the brotherhood, and also swore to French mathematician Augustin-
heights of pyramids and applied keep its mathematical discoveries Louis Cauchy completed the proof.
deductive reasoning to geometry. secret. Under Pythagoras’s
leadership, the community gained Fascinated by numbers
A Pythagorean community considerable political influence. Another type of number that
After 20 years of travelling, As well as his theorem, excited Pythagoras was the perfect
Pythagoras eventually settled in Pythagoras and his close-knit number. It was so called because it
Croton (now Crotone), southern community made numerous other is the exact sum of all the divisors
Italy, a city with a large Greek advances in mathematics, but less than itself. The first perfect
population. There, he established carefully guarded that knowledge. number is 6, as its divisors 1, 2,
the Pythagorean brotherhood – a Among their discoveries were and 3 add up to 6. The second is
community to whom he could polygonal numbers: these, when 28 (1 + 2 + 4 + 7 + 14 = 28), the
teach both his mathematical and represented by dots, can form the third 496, and the fourth 8,128.
philosophical beliefs. Women were shapes of regular polygons. For
welcome in the brotherhood, and example, 4 is a polygonal number
as 4 dots can form a square, and
10 is a polygonal number as 10 dots
can form a triangle with 4 dots at
the base, 3 dots on the next row,
2 on the next, and 1 dot at the top The finest type of man
of the triangle (4 + 3 + 2 + 1 = 10). gives himself up to
Strength of mind rests Two millennia after Pythagoras, discovering the meaning
in sobriety; for this keeps in 1638, Pierre de Fermat enlarged and purpose of life itself…
your reason unclouded on this idea when he asserted that this is the man I call a
by passion. any number could be written as the philosopher.
Pythagoras sum of up to k k-gonal numbers; Pythagoras
in other words, every single number
is the sum of up to 3 triangular
numbers, up to 4 square numbers,
or up to 5 pentagonal numbers,
and so on. For example, 19 can
 ANCIENT AND CLASSICAL PERIODS 41
shape was formed from 12 regular credited with coining the term
pentagons, and known as the “philosopher”, from the Greek philos
dodecahedron – one of the five (“love”) and sophos (“wisdom”).
Platonic solids. Pythagoreans For Pythagoras and his successors,
revered the pentagon, and their the duty of a philosopher was the
I have often admired the symbol was the pentagram, a pursuit of wisdom.
mystical way of Pythagoras, five-pointed star with a pentagon Pythagoras’s own brand of
and the secret at its centre. Breaking the philosophy integrated spiritual
magick of numbers. brotherhood’s rule of secrecy by ideas with mathematics, science,
Sir Thomas Browne revealing their knowledge of the and reasoning. Among his beliefs
English polymath dodecahedron would therefore have was the idea of metempsychosis,
been an especially heinous crime, which he may have encountered
punishable by death. on his travels in Egypt or elsewhere
in the Middle East. This held that
An integrated philosophy souls are immortal and at death
In ancient Greece, mathematics transmigrate to occupy a new
and philosophy were considered body. In Athens two centuries later,
There was no practical value in complementary subjects and were Plato was entranced by the idea
identifying such numbers, but their studied together. Pythagoras is and included it in many of his ❯❯
quirkiness and the beauty of their
patterns fascinated Pythagoras
and his brotherhood.
By contrast, Pythagoras was
said to have an overwhelming fear
and disbelief of irrational numbers,
numbers that cannot be expressed
as fractions of two integers, the
most famous example being p.
Such numbers had no place among
the well-ordered integers and
fractions by which Pythagoras
claimed the Universe was
governed. One story suggests
that his fear of irrational numbers
drove his followers to drown a
fellow Pythagorean – Hippasus –
for revealing their existence when
attempting to find √2.
Pythagoras’s reputation for
ruthlessness is also highlighted
in a story about a member of the
brotherhood who was executed
for publicly disclosing that the
Pythagoreans had discovered a
new regular polyhedron. The new

In The School of Athens, painted by

Raphael in 1509–11 for the Vatican in
Rome, Pythagoras is shown with a
book, surrounded by scholars eager
to learn from him.
42 PYTHAGORAS
and harmony. It is said that, while
walking past a blacksmith’s forge,
Pythagoras noticed that numerical patterns are found in
music and shapes. he noticed that different notes
were produced when hammers
of unequal weight were struck
against equal lengths of metal.
If the weights of the hammers
were in exact and particular
Some families The ratios of the A hammer proportions, their resulting notes
of numbers are lengths of lyre double the were harmonic.
polygonal; when strings are related to weight of a The hammers in the forge had
represented by the sequence of second hammer individual weights of 6, 8, 9, and
dots, they create notes in a will produce a note 12 units. Those weighing 6 and 12
regular polygons. musical scale. an octave lower.
units sounded the same notes at
different pitches; in today’s music
terminology they would be said to
be an octave apart. The frequency
of the note produced by the
Numbers and the ratios between them govern shapes and the hammer of weight 6 was double
sounds made by musical instruments and tools. that of the hammer weighing 12,
which corresponds with the ratio
of their weights. The hammers of
weights 12 and 9 produced a
dialogues. Later, Christianity, too, for entertainment. It was a unifying harmonious sound – a perfect
embraced the idea of a division element in his concept of Harmonia, fourth – as their weights were in
between body and soul; and the joining together of the cosmos the ratio 4:3. The notes made by the
Pythagoras’s ideas would become and the psyche. This may be why hammers of weights 12 and 8 were
a core part of Western thought. he is credited with discovering the also harmonious – a perfect fifth – as
Importantly for mathematics, link between mathematical ratios their weights were in the ratio 3:2.
Pythagoras also believed that
everything in the Universe related to
numbers and obeyed mathematical
rules. Certain numbers were
endowed with characteristics and
spiritual significance in what
amounted to a kind of number
worship, and Pythagoras and his
followers sought mathematical
patterns in everything around them.

Numbers in harmony
Music was of great importance
to Pythagoras. He is said to have
considered it a holy science, rather
than something simply to be used

Pythagoras was reputedly an

excellent lyre player. This drawing of
ancient Greek musicians illustrates
two members of the lyre family – the
trigonon (left) and the cithara.
 ANCIENT AND CLASSICAL PERIODS 43
The numerology of the Divine
Comedy by Dante (1265–1331) –
pictured here in a fresco from the
Duomo in Florence, Italy – reflects the
influence of Pythagoras, whom Dante
mentions several times in his writings.

Although the Pythagorean scale

worked well for music lying within
one octave, it was not suited for
more modern music, which was
written in different keys and
extended across several octaves.
While there have been many Pythagoras
In contrast, the hammers of different types of musical scales in
weights 9 and 8 were dissonant, use by different cultures, the long Pythagoras was born around
as 9:8 is not a simple mathematical tradition of Western music dates 570 bce on the Greek island of
ratio. By noticing that harmonious back to the Pythagoreans and Samos in the eastern Aegean
musical notes were connected to their quest to understand the Sea. His ideas have influenced
numerical ratios, Pythagoras was relationship between music and many of the greatest scholars
the first to uncover the relationship mathematical proportions. in history, from Plato to
between mathematics and music. Nicolaus Copernicus,
The legacy of Pythagoras Johannes Kepler, and Isaac
Creating a musical scale Pythagoras’s status as the most Newton. Pythagoras is
Although scholars have questioned famous mathematician from thought to have travelled
widely, assimilating ideas
the story of the forge, Pythagoras is antiquity is justified by his
from scholars in Egypt and
also widely credited with another contributions to geometry, number
elsewhere in the Middle
musical discovery. He is said to theory, and music. His ideas were East before establishing his
have experimented with notes not always original, but the rigour community of around 600
produced by lyre strings of different with which he and his followers people in Croton, southern
lengths. He found that while a developed them, using axioms Italy, around 518 bce. This
vibrating string produces a note and logic to build a system of ascetic brotherhood required
with frequency f, halving the length mathematics, was a fine legacy its members to live for
of the string produces a note an for those who succeeded him. ■ intellectual pursuits, while
octave higher, with frequency following strict rules of diet
2f. When Pythagoras used the and clothing. It is from this
same ratios that produced time onwards that his theorem
harmoniously sounding hammers, and other discoveries were
and applied them to vibrating probably set down, although
strings, he similarly produced no records remain. At the age
There is geometry of 60, Pythagoras is said to
notes in harmony with one another.
in the humming of have married a young member
Pythagoras then constructed a of the community, Theano,
musical scale, starting with one the strings, there is music and perhaps had two or three
note and the note an octave above in the spacing of children. Political upheaval in
it, filling in the notes between the spheres. Croton led to a revolt against
using perfect fifths. Pythagoras the Pythagoreans. Pythagoras
This scale was used until the may have been killed when
16th century, when it was replaced his school was set on fire, or
by the even-tempered scale, in shortly afterwards. He is said
which the notes between the two to have died around 495 bce.
octaves are more evenly spaced.
44

A REAL NUMBER
THAT IS NOT
RATIONAL
IRRATIONAL NUMBERS

A
ny number that can be
IN CONTEXT expressed as a ratio of
two integers – a fraction,
KEY FIGURE
a decimal that either ends or

ram
Hippasus (5th century bce)
repeats in a recurring pattern,

ntag
FIELD or a percentage – is said to be a

of pe
Number systems rational number. All whole numbers
are rational as they can be shown

Side
BEFORE as fractions divided by 1. Irrational
19th century bce Cuneiform numbers, however, cannot be
inscriptions show that the expressed as a ratio of two numbers
Babylonians constructed Hippasus, a Greek scholar, is
right-angled triangles and believed to have first identified Side of pentagon
understood their properties. irrational numbers in the 5th Hippasus may have encountered
century bce, as he worked on irrational numbers while exploring the
6th century bce In Greece, geometrical problems. He was relationship between the length of the
the relationship between the familiar with Pythagoras’s theorem, side of a pentagon and one side of a
side lengths of a right-angled which states that the square of pentagram formed inside it. He found
triangle is discovered, and is that it was impossible to express it as
the hypotenuse in a right-angled a ratio between two whole numbers.
later attributed to Pythagoras. triangle is equal to the sum of the
AFTER squares of the other two sides.
400 bce Theodorus of Cyrene He applied the theorem to a right- root of 2 an irrational number, and
angled triangle that has both shorter 2 itself is termed non-square or
proves the irrationality of the
sides equal to 1. As 12 + 12 = 2, the square-free. The numbers 3, 5, 7,
square roots of the non-square
length of the hypotenuse is the and many others are similarly
numbers between 3 and 17. square root of 2. non-square and in each case, their
4th century bce The Greek Hippasus realized, however, that square root is irrational. By contrast,
mathematician Eudoxus of the square root of 2 could not be numbers such as 4 (22), 9 (32), and
Cnidus establishes a strong expressed as the ratio of two whole 16 (42) are square numbers, with
mathematical foundation for numbers – that is, it could not be square roots that are also whole
irrational numbers. written as a fraction, as there is no numbers and therefore rational.
rational number that can be The concept of irrational
multiplied by itself to produce numbers was not readily accepted,
precisely 2. This makes the square although later Greek and Indian
 ANCIENT AND CLASSICAL PERIODS 45
See also: Positional numbers 22–27 ■ Quadratic equations 28–31 ■ Pythagoras
36–43 ■ Imaginary and complex numbers 128–31 ■ Euler’s number 186–91

An irrational number has

A real number gives a
an infinite number
positive result when
of decimals with
it is squared.
no recurring pattern.

Hippasus
The square root of 2 is
The square root of 2 gives a
1.14142135… where the Details of Hippasus’s early
positive result – 2 – when
decimals continue with no life are sketchy, but it is
it is squared.
recurring pattern. thought that he was born in
Metapontum, in Magna
Graecia (now southern Italy),
around 500 bce. According to
the philosopher Iamblichus,
who wrote a biography of
The square root of 2 is a real number Pythagoras, Hippasus was a
that is not rational. founder of a Pythagorean sect
called the Mathematici, which
fervently believed that all
numbers were rational.
mathematicians explored their rational, as will the average of that Hippasus is usually
properties. In the 9th century, Arab number and either of the original credited with discovering
scholars used them in algebra. numbers. Irrational numbers can irrational numbers, an idea
also be found between any two that would have been
In decimal terms rational numbers. One method is considered heresy by the
The positional decimal system of to change a digit in a recurring sect. According to one story,
Hindu–Arabic numeration allowed sequence. For example, an irrational Hippasus drowned when his
further study of irrational numbers, number can be found between the fellow Pythagoreans threw
which can be shown as an infinite recurring numbers 0.124124… and him over the side of a boat
series of digits after the decimal 0.125125… by changing 1 to 3 in in disgust. Another story
point with no recurring pattern. For the second cycle of 124, to give suggests that a fellow
example, 0.1010010001… with an 0.124324…, and doing so again at Pythagorean discovered
extra zero between each successive the fifth, then ninth cycle, increasing irrational numbers, but
Hippasus was punished for
pair of 1s, continuing indefinitely, is the gap between the replacement
telling the outside world about
an irrational number. Pi (), which 3s by one cycle each time. them. The year of Hippasus’s
is the ratio of the circumference of One of the great challenges of death is not known but is
a circle to its diameter, is irrational. modern number theory has been likely to have been in the
This was proved in 1761 by Johann establishing whether there are 5th century bce.
Heinrich Lambert – earlier more rational or irrational numbers.
estimations of  had been 3 or 22 ⁄7. Set theory strongly indicates that Key works
Between any two rational there are many more irrational
numbers, another rational number numbers than rational numbers, 5th century bce Mystic
can always be found. The average even though there are infinite Discourse
of the two numbers will also be numbers of each. ■
46

THE QUICKEST
RUNNER CAN
NEVER OVERTAKE
THE SLOWEST
ZENO’S PARADOXES OF MOTION

Z
eno of Elea belonged to the absurdity of the pluralist view that
IN CONTEXT Eleatic school of philosophy motion can be divided. A body
that flourished in ancient moving a certain distance, it says,
KEY FIGURE
Greece in the 5th century bce. In would have to reach the halfway
Zeno of Elea (c. 495–430 bce)
contrast to the pluralists, who point before it arrived at the end,
FIELD believed that the Universe could be and in order to reach that halfway
Logic divided into its constituent atoms, mark, it would first have to reach
Eleatics believed in the indivisibility the quarter-way mark, and so on ad
BEFORE of all things. infinitum. Because the body has to
Early 5th century bce The Zeno wrote 40 paradoxes to pass through an infinite number of
Greek philosopher Parmenides show the absurdity of the pluralist points, it would never reach its goal.
founds the Eleatic school of view. Four of these – the dichotomy In the paradox of Achilles and
philosophy in Elea, a Greek paradox, Achilles and the tortoise, the tortoise, Achilles, who is 100
colony in southern Italy. the arrow paradox, and the stadium times faster than the tortoise,
paradox – address motion. The gives the creature a head start of
AFTER dichotomy paradox shows the 100 metres in a race. At the sound
350 bce Aristotle produces
his treatise Physics, in which
he draws on the concept
of relative motion to refute
Zeno’s paradoxes. In Zeno’s arrow At any given time, this
paradox, an arrow arrow occupies a static
1914 British philosopher point in space.
is fired.
Bertrand Russell, who
described Zeno’s paradoxes as
immeasurably subtle, states
that motion is a function of
position with respect to time.

The arrow is frozen in

The flying arrow
stasis at every moment
is stationary.
in its flight.
 ANCIENT AND CLASSICAL PERIODS 47
See also: Pythagoras 36–43 ■ Syllogistic logic 50–51 ■ Calculus 168–75 ■ Transfinite numbers 252–53 ■ The logic of
mathematics 272–73 ■ The infinite monkey theorem 278–79

of the starting signal, Achilles runs

100 metres to reach the tortoise’s
starting point, while the tortoise
runs 1 metre, giving it a 1 metre
lead. Undeterred, Achilles runs
another metre; however, in the 100m Starting point of 101m
same time, the tortoise runs one- tortoise and end point
of Achilles
hundredth of a metre, so it is still
in the lead. This continues, and
Achilles never catches up.
The stadium paradox concerns
three columns of people, each
containing an equal number of
101m 101.01m
people; one group is at rest, while
the other two run past each other
at the same speed in opposite
directions. According to the
paradox, a person in one moving
group can pass two people in the
other moving group in a fixed time,
but only one person in the stationary
101.01m 101.0101m
group. The paradoxical conclusion is
that half a given time is equivalent The paradox of Achilles and
to double that time. the tortoise maintains that a
Over the centuries, many fast object, such as Achilles,
will never catch up with a
mathematicians have refuted the slow one, such as a tortoise.
paradoxes. The development of Achilles will get closer to the
calculus allowed mathematicians to tortoise, but never actually
deal with infinitesimal quantities overtake it. 101.0101m 101.010101m
without resulting in contradiction. ■

Zeno of Elea Zeno of Elea was born around him as the inventor of the
495 bce in the Greek city of Elea dialectical method (a method
(now Velia, in southern Italy). At a starting from two opposing
young age, he was adopted by the viewpoints) of logical argument.
philosopher Parmenides, and was Zeno collected his arguments in
said to have been “beloved” by a book, but this did not survive.
him. Zeno was inducted into the The paradoxes are known from
school of Eleatic thought, founded Aristotle’s treatise Physics,
by Parmenides. At the age of which lists nine of them.
around 40, Zeno travelled to Although little is known of
Athens, where he met Socrates. Zeno’s life, the ancient Greek
Zeno introduced the Socratic biographer Diogenes claimed he
philosophers to Eleatic ideas. was beaten to death for trying to
Zeno was renowned for his overthrow the tyrant Nearchus.
paradoxes, which contributed to In a clash with Nearchus, Zeno
the development of mathematical is reported to have bitten off the
rigour. Aristotle later described man’s ear.
48

THEIR COMBINATIONS
GIVE RISE TO
ENDLESS
COMPLEXITIES
THE PLATONIC SOLIDS

T
he perfect symmetry of
IN CONTEXT the five Platonic solids was
A regular polygon has probably known to scholars
KEY FIGURE
equal angles long before the Greek philosopher
Plato (c. 428–348 bce)
and equal sides. Plato popularized the forms in his
FIELD dialogue Timaeus, written in
Geometry c. 360 bce. Each of the five regular
convex polyhedra – 3-D shapes with
BEFORE flat faces and straight edges – has
6th century bce Pythagoras its own set of identical polygonal
identifies the tetrahedron, faces, the same number of faces
cube, and dodecahedron. Only five solids (3-D shapes)
meeting at each vertex, as well as
have identical vertices and
faces that are all identical equilateral sides, and same-sized
4th century bce Theaetetus,
regular polygons. angles. Theorizing on the nature of
an Athenian contemporary
the world, Plato assigned four of the
of Plato, discusses the shapes to the classical elements:
octahedron and icosahedron. the cube (also known as a regular
AFTER hexahedron) was associated with
c. 300 bce Euclid’s Elements earth; the icosahedron with water;
fully describes the five regular These five solids the octahedron with air; and the
are the tetrahedron, tetrahedron with fire. The 12-faced
convex polyhedra.
cube, octahedron, dodecahedron was associated with
1596 German astronomer dodecahedron, and the heavens and its constellations.
Johannes Kepler proposes icosahedron.
a model of the Solar System, Composed of polygons
explaining it geometrically Only five regular polyhedra are
in terms of Platonic solids. possible – each one created either
from identical equilateral triangles,
1735 Leonhard Euler devises squares, or regular pentagons, as
a formula that links the Euclid explained in Book XIII of
faces, vertices, and edges They are known as the his Elements. To create a Platonic
of polyhedra. Platonic solids.
solid, a minimum of three identical
polygons must meet at a vertex,
so the simplest is a tetrahedron –
 ANCIENT AND CLASSICAL PERIODS 49
See also: Pythagoras 36–43 ■ Euclid’s Elements 52–57 ■ Conic sections 68–69 ■ Trigonometry 70–75
■ Non-Euclidean geometries 228–29 ■ Topology 256–59 ■ The Penrose tile 305

The Platonic solids

A tetrahedron has A cube has six An octahedron has A dodecahedron has An icosahedron has
four triangular faces. square faces. eight triangular faces. 12 pentagonal faces. 20 triangular faces.

a pyramid made up of four and 20 vertices), and an icosahedron acknowledged he was wrong, but
equilateral triangles. Octahedra (20 faces and 12 vertices) form his calculations led him to discover
and icosahedra are also formed another dual pair. Tetrahedra, which that planets have elliptical orbits.
with equilateral triangles, while have four faces and four vertices, In 1735, Swiss mathematician
cubes are created from squares, are said to be self-dual. Leonhard Euler noted a further
and dodecahedra are constructed property of Platonic solids, later
with regular pentagons. Shapes in the Universe? shown to be true for all polyhedra.
Platonic solids also display Like Plato, later scholars sought The sum of the vertices (V ) minus
duality: the vertices of one Platonic solids in nature and the the number of edges ( E) plus the
polyhedron correspond to the faces Universe. In 1596, Johannes Kepler number of faces ( F) always equals
of another. For example, a cube, reasoned that the positions of the 2, that is, V  E + F = 2.
which has six faces and eight six planets then known (Mercury, It is also now known that Platonic
vertices, and an octahedron (eight Venus, Earth, Mars, Jupiter, and solids are indeed found in nature – in
faces and six vertices) form a dual Saturn) could be explained in terms certain crystals, viruses, gases, and
pair. A dodecahedron (12 faces of the Platonic solids. Kepler later the clustering of galaxies. ■

Plato Born around 428 bce to wealthy geometry, believing that its
Athenian parents, Plato was a forms – especially the five
student of Socrates, who was regular convex polyhedra –
also a family friend. Socrates’ could explain the properties
execution in 399 bce deeply of the Universe. Plato found
affected Plato and he left Greece perfection in mathematical
to travel. During this period objects, believing they were
his discovery of the work of the key to understanding the
Pythagoras inspired a love of differences between the real
mathematics. Returning to and the abstract. He died in
Athens, in 387 bce he founded Athens around 348 bce.
the Academy, inscribing over its
entrance the words “Let no Key works
one ignorant of geometry enter
here”. Teaching mathematics c. 375 bce The Republic
as a branch of philosophy, Plato c. 360 bce Philebus
emphasized the importance of c. 360 bce Timaeus
50

DEMONSTRATIVE
KNOWLEDGE MUST
REST ON NECESSARY
BASIC TRUTHS
SYLLOGISTIC LOGIC

I
n Classical Greece, there was model entirely satisfactory, so he
IN CONTEXT no clear distinction between set about determining a systematic
mathematics and philosophy; structure for logical argument.
KEY FIGURE
the two were considered First, he identified the different
Aristotle (384–322 bce)
interdependent. For philosophers, kinds of proposition that can be
FIELD one important principle was the used in logical arguments, and how
Logic formulation of cogent arguments they can be combined to reach a
that followed a logical progression logical conclusion. In Prior Analytics,
BEFORE of ideas. The principle was based he describes the propositions as
6th century bce Pythagoras on Socrates’ dialectal method of being of broadly four types, in the
and his followers develop a questioning assumptions to expose form of “all S are P”, “no S are P”,
systematic method of proof inconsistencies and contradictions. “some S are P”, and “some S are not
for geometric theorems. Aristotle, however, did not find this P”, where S is a subject, such as
AFTER
c. 300 bce Euclid’s Elements ALL S ARE P NO S ARE P
describes geometry in terms of universal affirmative universal negative
logical deduction from axioms. A Contrary propositions E

1677 Gottfried Leibniz

suggests a form of symbolic
notation for logic, anticipating Subalternate Contradictory Subalternate
the development of propositions propositions propositions
mathematical logic.
1854 George Boole publishes
The Laws of Thought, his I Subcontrary propositions O
second book on algebraic logic. particular affirmative particular negative
SOME S ARE P SOME S ARE NOT P
1884 The Foundations of
Arithmetic by German In the Square of Opposition, S is a subject, such as “sugar”, and P a predicate,
mathematician Gottlob Frege such as “sweet”. A and O are contradictory, as are E and I (if one is true, the other
is false, and vice versa). A and E are contrary (both cannot be true but both can
examines the logical principles be false); I and O are subcontrary: both can be true but both cannot be false. I is
underpinning mathematics. a subaltern of A and O is a subaltern of E. In syllogistic logic, this means that if
A is true, I must be true, but that if I is false, A must be false as well.
 ANCIENT AND CLASSICAL PERIODS 51
See also: Pythagoras 36–43 ■ Zeno’s paradoxes of motion 46–47 ■ Euclid’s
Elements 52–57 ■ Boolean algebra 242–47 ■ The logic of mathematics 272–73

Syllogistic arguments start

with a major premise, which
All men are mortal.
is a universal or
general rule.

This then leads to a

particular case, or a Aristotle is a man.
minor premise. Aristotle
The son of a physician at the
Macedonian court, Aristotle
was born in 384 bce, in Stagira,
Chalkidiki. At the age of about
The conclusion is drawn 17, he left to study at Plato’s
from these major and Aristotle is mortal. Academy in Athens, where
minor premises. he excelled. Soon after Plato’s
death, anti-Macedonian
prejudice forced him to leave
Athens. He continued his
academic work in Assos (now
Syllogistic logic follows the same process of in Turkey). In 343 bce, Philip II
recalled him to Macedonia
deduction as mathematical proofs. to head the school at the court;
one of his students was
Philip’s son, later known as
sugar, and P the predicate – a rule in the major premise, such Alexander the Great.
quality, such as sweet. From just as “All men are mortal”, and a In 335 bce, Aristotle
two such propositions an argument particular case in the minor returned to Athens and
can be constructed and a conclusion premise, such as “Aristotle is a founded the Lyceum, a rival
deduced. This is, in essence, man”, to reach a conclusion that institution to the Academy.
the logical form known as the necessarily follows – in this case, In 323 bce, after Alexander’s
syllogism: two premises leading “Aristotle is mortal”. This form death, Athens again became
to a conclusion. Aristotle identified of deductive reasoning is the fiercely anti-Macedonian, and
the structure of syllogisms that foundation of mathematical proofs. Aristotle retired to his family
estate in Chalcis, on Euboea.
are logically valid, those where Aristotle notes in Posterior
He died there in 322 bce.
the conclusion follows from the Analytics that, even in a valid
premises, and those that are not, syllogistic argument, a conclusion
where the conclusion does not cannot be true unless it is based Key works
follow from the premises, providing on premises accepted as true,
a method for both constructing such as self-evident truths or c. 350bce Prior Analytics
and analysing logical arguments. axioms. With this idea, he c. 350 bce Posterior Analytics
established the principle of c. 350 bce On Interpretation
Seeking a rigorous proof axiomatic truths as the basis for 335–323 bce Nichomachean
Implicit in his discussion of valid a logical progression of ideas – the Ethics
syllogistic logic is the process of model for mathematical theorems 335–323bce Politics
deduction, working from a general from Euclid onwards. ■
THE WHOLE IS
GREATER
THAN THE PART
EUCLID’S ELEMENTS
54 EUCLID’S ELEMENTS

E
uclid’s Elements has a
IN CONTEXT strong claim for being
the most influential
KEY FIGURE
mathematical work of all time.
Euclid (c. 300 bce)
It dominated human conceptions
FIELD of space and number for more than There is
Geometry 2,000 years and was the standard no royal road
geometrical textbook until the to geometry.
BEFORE start of the 20th century. Euclid
c. 600 bce The Greek Euclid lived in Alexandria,
philosopher, mathematician, Egypt, in around 300 bce, when
and astronomer Thales of the city was part of the culturally
Miletus deduces that the rich Greek-speaking Hellenistic
angle inscribed inside a world that flourished around the
semicircle is a right angle. Mediterranean Sea. He would
This becomes Proposition have written on papyrus, which are devoted to number theory
31 of Euclid’s Elements. is not very durable; all that and discuss the properties and
remains of his work are the copies, relationships of numbers. The
c. 440 bce The Greek translations, and commentaries long and difficult Book X deals with
mathematician Hippocrates made by later scholars. incommensurables. Now known as
of Chios writes the first irrational numbers, these numbers
systematically organized Collection of works cannot be expressed as a ratio of
geometry textbook, Elements. The Elements is a collection of integers. Books XI to XIII examine
13 books that range widely in three-dimensional solid geometry.
AFTER
subject matter. Books I to IV tackle Book XIII of the Elements
c. 1820 Mathematicians plane geometry – the study of is actually attributed to another
such as Carl Friedrich Gauss, flat surfaces. Book V addresses author – Athenian mathematician
János Bolyai, and Nicolai the idea of ratio and proportion, and disciple of Plato, Theaetetus,
Ivanovich Lobachevsky begin inspired by the thinking of the who died in 369 bce. It covers the
to move towards hyperbolic Greek mathematician and five regular convex solids – the
non-Euclidean geometry. astronomer Eudoxus of Cnidus. tetrahedron, cube, octahedron,
Book VI contains more advanced dodecahedron, and icosahedron,
plane geometry. Books VII to IX which are often called the Platonic

Euclid Details of Euclid’s date and astronomy, number theory, and

place of birth are unknown and the importance of mathematical
knowledge of his life is scant. It rigour. Several of the works
is thought that he studied at the attributed to Euclid have been
Academy in Athens, which had lost, but at least five have
been founded by Plato. In the 5th survived to the 21st century.
century ce, the Greek philosopher It is thought that Euclid died
Proclus wrote in his history of between the mid-4th century
mathematicians that Euclid taught and the mid-3rd century bce.
at Alexandria during the reign of
Ptolemy I Soter (323–285 bce). Key works
Euclid’s work covers two
areas: elementary geometry and Elements
general mathematics. In addition Conics
to the Elements, he wrote about Catoptrics
perspective, conic sections, Phaenomena
spherical geometry, mathematical Optics
 ANCIENT AND CLASSICAL PERIODS 55
See also: Pythagoras 36–43 ■ The Platonic solids 48–49 ■ Syllogistic logic 50–51 ■ Conic sections 68–69 ■ The problem
of maxima 142–43 ■ Non-Euclidean geometries 228–29

solids – and is the first recorded element as “a letter of an alphabet”,

example of a classification theorem with combinations of letters
(one that itemizes all possible creating words in the same way
figures given certain limitations). that combinations of axioms –
Euclid is known to have written statements that are self-evidently
an account of conic sections, but true – create propositions.
this work has not survived. Conic
sections are figures formed from Logical deductions
the intersection of a plane and a Euclid was not writing in a
cone and they may be circular, vacuum; he built upon foundations
elliptical, or parabolic in shape. laid by a number of influential
Greek mathematicians who came
World of proof before him. Thales of Miletus,
The title of Euclid’s work has a Hippocrates, and Plato (among
particular meaning that reflects others) had all begun to move
his mathematical approach. In the towards the mathematical mindset
20th century, British mathematician that Euclid so brilliantly formalized:
This opening page of Euclid’s
John Fauvel maintained that the the world of proof. It is this that Elements shows illuminated Latin
meaning of the Greek word for makes Euclid unique; his writings text with diagrams and comes from
“element”, stoicheia, changed over are the earliest surviving example the first printed edition, produced in
time, from “a constituent of a line”, of fully axiomatized mathematics. Venice in 1482.
such as an olive tree in a line of He identified certain basic facts
trees, to “a proposition used to and progressed from there to
prove another”, and eventually statements that were sound logical logical relationships between
evolved to mean “a starting point deductions (propositions). Euclid the various propositions were
for many other theorems”. This also managed to assemble all carefully explained.
is the sense in which Euclid used the mathematical knowledge of Euclid faced a Herculean task
it. In the 5th century ce, the his day, and organize it into a when he attempted to systematize
philosopher Proclus talked of an mathematical structure where the the mathematics that lay before
him. In devising his axiomatic
system, he began with 23 definitions
for terms such as point, line,
The Euclidean axiomatic surface, circle, and diameter. He
system holds that… then put forward five postulates:
any two points can be joined
with a straight line segment;
any straight line segment can be
extended to infinity; given any
axioms, or postulates are straight line segment, a circle can
common notions, statements about be drawn having the segment as
are assumed to be geometry that are
its radius and one endpoint as its
universally true. assumed to be true.
centre; all right angles are equal to
one another; and a postulate about
parallel lines (see p.56).
He then went on to add five
All theorems are derived from axioms axioms, or common notions; if A =
and postulates. B and B = C, then A = C; if A = B
and C = D, A + C = B + D; ❯❯
56 EUCLID’S ELEMENTS
Euclid’s five postulates

1. Any two points can be x

joined with a straight line
segment.
y

2. Any straight line segment 3. Given a centre and a 4. All right angles 5. If x + y is less than two right
can be extended to infinity. radius, a circle can always are equal to one angles, then the lines must
be drawn with this centre another. eventually meet on one side.
and this radius.

if A = B and C = D, then A - C = same, so AC = AB and BC = AB; definitions describes a triangle as a

B - D; if A coincides with B, then this means that AC = BC, which plane figure bounded by three lines,
A and B are equal; and the whole of is Euclid’s first axiom (things that which all lie in that plane. However,
A is greater than part of A. are equal to the same thing are it seems that Euclid did not
To prove Proposition 1 (see also equal to one another). It follows explicitly show that the lines AB,
opposite), Euclid drew a line with that AB = BC = CA, meaning that BC, and CA lie in the same plane.
endpoints labelled A and B (see he had drawn an equilateral Postulate 5 is also known as
below). Taking each endpoint triangle on AB. the “parallel postulate” because
as a centre, he then drew two In Latin translations of Elements, it can be used to prove properties
intersecting circles, so that each deductions end with the letters of parallel lines. It says that if a
had the radius AB. This used his QEF (quod erat faciendum, meaning straight line crossing two straight
third postulate. Where the circles “which was to be [and has been] lines ( A, B) creates interior angles
met, he called that point C, and he done”. Logical proofs end with on one side that total less than two
could draw two more lines AC and QED (quod erat demonstrandum, right angles (180°), lines A and B
BC, calling on his first postulate. meaning “which was to be [and will eventually cross on that side,
The radius of the two circles is the has been] demonstrated”). if extended indefinitely. Euclid did
The equilateral triangle not use it until Proposition 29, in
construction is a good example which he stated that one condition
C of Euclid’s method. Each step has for a straight line crossing two
to be justified by reference to the parallel lines was that the interior
definitions, the postulates, and the
axioms. Nothing else can be taken
as obvious, and intuition is
A B regarded as potentially suspect.
Euclid’s very first proposition
was criticized by later writers. They
noted, for instance, that Euclid did Geometry is
not justify or explain the existence knowledge of what
of C, the point of intersection of the always exists.
To construct an equilateral triangle, two circles. Although apparent, it is Plato
for Proposition 1, Euclid drew a line and not mentioned in his preliminary
centred a circle on each of its endpoints,
here A and B. By drawing a line from assumptions. Postulate 5 talks
each endpoint to C, where the circles about a point of intersection, but
intersect, he created a triangle with that is between two lines, and not
sides AB, AC, and BC of equal length. two circles. Similarly, one of the
 ANCIENT AND CLASSICAL PERIODS 57
angles on the same side were
equal to two right angles. The fifth The first 16 propositions in Book 1
postulate is more elaborate than Proposition 1 On a given finite straight line, to construct an equilateral triangle.
the other four, and Euclid himself
Proposition 2 To place at a given point (as an extremity) a straight line equal
seems to have been wary of it. to a given straight line.
A vital part of any axiomatic
system is to have enough axioms, Proposition 3 Given two unequal straight lines, to cut off from the greater a
and postulates in the case of Euclid, straight line equal to the less.
to derive every true proposition,
Proposition 4 If two sides of one triangle are equal in length to two sides of
but to avoid superfluous axioms another triangle, and if the angles contained by each pair of
that can be derived from others. equal sides are equal, then the base of one triangle will equal
Some asked whether the parallel the base of the other, the two triangles will be of equal area,
and the remaining angles in one triangle will be equal to those
postulate could be proved as a in the other triangle.
proposition using Euclid’s common
notions, definitions, and the other Proposition 5 In an isosceles triangle, the angles at the base are equal to one
four postulates; if it could, the another, and, if the equal straight lines are extended below the
base, the angles under the base will also be equal to one another.
fifth was unnecessary. Euclid’s
contemporaries and later scholars Proposition 6 If in a triangle two angles are equal to one another, the sides
made unsuccessful attempts to separated from the third side by these angles will also be equal.
construct such a proof. Finally, in
the 19th century, the fifth postulate Proposition 7 Given two straight lines constructed on a straight line (from its
extremities) and meeting in a point, there cannot be constructed
was ruled both necessary for on the same straight line (from its extremities), and on the same
Euclid’s geometry and independent side of it, two other straight lines meeting in another point and
of his other four postulates. equal to the former two respectively, namely each to that which
starts at the same extremity.
Beyond Euclidean geometry
Proposition 8 If two sides of one triangle are equal in length to two sides of
The Elements also examines another triangle, and the base of one triangle is equal to the base
spherical geometry, an area explored of the other, the angles of the two triangles will also be equal.
by two of Euclid’s successors,
Theodosius of Bithynia and
Proposition 9 To bisect a given rectilineal angle.
Menelaus of Alexandria. While
Euclid’s definition of “a point” Proposition 10 To bisect a given finite straight line.
addresses a point on the plane, a
point can also be understood as Proposition 11 To draw a straight line at right angles to a given straight line
a point on a sphere. from a given point on it.
This raises the question of how
Euclid’s five postulates can be Proposition 12 To a given infinite straight line, from a given point which is not
on it, to draw a perpendicular straight line.
applied to the sphere. In spherical
geometry, almost all the axioms Proposition 13 If a straight line set up on a straight line makes angles, it will
look different from the postulates make either two right angles or angles equal to two right angles.
set out in Euclid’s Elements. The
Elements gave rise to what is called Proposition 14 If with any straight line, and at a point on it, two straight lines
Euclidean geometry; spherical not lying on the same side and meeting at the point make
geometry is the first example of adjacent angles equal to two right angles, the two straight
lines will be in a straight line with one another.
a non-Euclidean geometry. The
parallel postulate is not true for Proposition 15 If two straight lines cut one another, they make the vertical
spherical geometry, where all angles equal to one another.
pairs of lines have points in
common, nor for hyperbolic Proposition 16 In any triangle, if one of the sides is extended, the angle between
the triangle and the extended side is greater than any of the
geometry, where they can meet angles inside the triangle.
infinite numbers of times. ■
58

COUNTING
WITHOUT
NUMBERS
THE ABACUS

T
he abacus is a counting drawing board. The oldest surviving
IN CONTEXT device and calculator that abacus is the Salamis Tablet, a
has been in use since marble slab made c. 300 bce that is
KEY CIVILIZATION
ancient times. It comes in many etched with horizontal lines. Pebbles
Ancient Greeks (c. 300 bce)
forms, but all of them work on the were placed on these lines to count
FIELD same principles: values of different out values. The bottom line
Number systems sizes are represented by “counters” represented 0 to 4; the line above
arranged in columns or rows. counted 5s, and the lines above that
BEFORE 10s, 50s, and so on. The tablet was
c. 18,000 bce In Central Africa, Early abaci discovered on the Greek island of
numbers are recorded on bone The word “abacus” may hint at its Salamis in 1846.
as carved marks. origins. It is a Latin word derived Some scholars believe that
from the ancient Greek, abax, the Salamis Tablet was actually
c. 3000 bce South American
which means “slab” or “board” – Babylonian. The Greek abax may
Indians record numbers by a surface that would have been have come from the Phoenician or
tying knots in string. covered in sand and used as a Hebrew word for “dust” (abaq) and
c. 2000 bce The Babylonians
develop positional numbers. The Soroban Championship
AFTER Japanese schoolchildren still accuracy in a knockout system
1202 Leonardo of Pisa use the soroban (Japanese similar to a spelling bee. One
(Fibonacci) commends the abacus) in mathematics lessons of the highlights of the event
Hindu–Arabic number system as a way of developing mental is Flash Anzan™, a feat of
in Liber Abaci. arithmetic skills. The soroban is mental arithmetic in which the
also used for far more complex players imagine operating an
1621 In England, William calculations. Expert soroban abacus to add 15 three-digit
Oughtred invents the slide users can usually do such numbers – no physical abacus is
rule, which simplifies the use calculations more quickly than allowed. The contestants watch
of logarithms. someone punching the values the numbers appear on a big
into an electronic calculator. screen, flashing by faster with
1972 Hewlett Packard invents Every year, the best abacists each round. The 2017 world
an electronic scientific from across Japan take part in record for Flash Anzan was
calculator for personal use. the Soroban Championship. They 15 numbers added together
are tested on their speed and in 1.68 seconds.
 ANCIENT AND CLASSICAL PERIODS 59
See also: Positional numbers 22–27 ■ Pythagoras 36–43 ■ Zero 88–91 ■ Decimals 132–37 ■ Calculus 168–75

The suanpan shown here is Ten millions Hundred thousands Ten thousands Thousands
set to the number 917,470,346.
The suanpan is traditionally a 2:5 Hundred Millions Hundreds Tens
millions Ones
abacus – each column has two
“heaven” beads, each with a
value of 5, and 5 “earth” beads, Heaven:
each with a value of 1, giving a 5-unit
potential value of 15 units. This counters
allows for calculations involving 5 5 5 5
the Chinese base-16 system,
which uses 15 units rather than 1 1 1 1 1 1 1 1
the 9 used in the decimal system.
Numbers can be added together 1 1 1 1 1 1
by entering the units of one 1 1 1 1
number, starting from the right, Earth:
then adjusting the beads as 1 1 1 1-unit
further numbers are entered. For counters
subtraction, the units of the first
number are entered, then bead
values are adjusted downwards
in each column as further
subtracted numbers are entered. 9 1 7, 4 7 0, 3 4 6

may refer to far older counting nepohualtzintzin – the “personal By the second millennium ce, the
tables developed in Mesopotamian accounts counter” – and wore it suanpan and its counting methods
civilizations, where counters were on the wrist as a bracelet. were becoming widespread across
set out on grids drawn in sand. Asia. In the 1300s, it was exported
The Babylonian positional number Double base to Japan, where it was called the
system, developed c. 2000 bce, may Around the 2nd century ce, abaci soroban. This was slowly refined
have been inspired by the abacus. had become a common tool in and by the 20th century, the soroban
The Romans upgraded the China. The Chinese abacus, or was a 1:4 abacus (with 1 upper bead
Greek counting table into a device suanpan, matched the design of the on each rod, and 4 lower beads). ■
that greatly simplified calculations. Roman version, but rather than use
The horizontal rows of the Greek pebbles set in a metal frame, it
abacus became vertical columns in employed wooden counters on
the Roman abacus, in which were rods – the template for modern abaci.
set small pebbles – or calculi in Whether the Roman or Chinese
Latin, from which we get the abaci came first is unclear, but their
word “calculation”. similarities may be a coincidence,
A type of abacus was also in use inspired by the way people count
in the pre-Columbian civilizations using the five fingers of one hand.
of Central America. Based on a Both abaci have two decks – the
five-digit vigesimal, or base-20, lower deck counting to five, and
counting system, it used maize the upper deck counting the fives.
kernels threaded on strings to
represent numbers. No device has
A female personification of
survived, but scholars think that Arithmetic judges a contest between
the ancient Olmec people invented the Roman mathematician Boëthius,
it 3,000 years ago. By about 1000 ce, who uses numbers, and the Greek
the Aztec people knew it as the Pythagoras, who uses a counting board.
EXPLORING PI
IS LIKE EXPLORING THE
UNIVERSE
CALCULATING PI
62 CALCULATING PI

T
he fact that pi () – the
IN CONTEXT ratio of the circumference
of a circle to its diameter,
KEY FIGURE
roughly given as 3.141 – cannot be
Archimedes
expressed exactly as a decimal no
(c. 287–c. 212 bce) matter how many decimal places Pi is not merely the
FIELD are calculated has fascinated ubiquitous factor in high
Number theory mathematicians for centuries. school geometry problems; it
Welsh mathematician William is stitched across the whole
BEFORE Jones was the first to use the tapestry of mathematics.
c. 1650 bce The Rhind Greek letter  to represent the Robert Kanigel
papyrus, written by Middle number in 1706, but its importance American science writer
Kingdom Egyptian scribes as for calculating the circumference
a mathematics guide, includes and area of a circle and the volume
estimates of the value of . of a sphere has been understood
for millennia.
AFTER
5th century ce In China, Ancient texts
Zu Chongzhi calculates  to Determining pi’s exact value is approximately 3.1605 calculated to
seven decimal places. not straightforward and the quest four decimal places, which is just
1671 Scottish mathematician continues to find pi’s decimal 0.6 per cent greater than the most
James Gregory develops the representation to as many places accurate known value of .
as possible. Two of the earliest In ancient Babylon, the area of a
arctangent method for
estimates for  are given in the circle was found by multiplying the
computing . Gottfried
ancient Egyptian documents square of the circumference by 1 ⁄12,
Leibniz makes the same
known as the Rhind and Moscow implying that the value of  was 3.
discovery in Germany three papyri. The Rhind papyrus, thought This value appears in the Bible
years later. to have been intended for trainee (1 Kings 7:23): “And he made the Sea
2019 In Japan, Emma Haruka scribes, describes how to calculate of cast bronze, ten cubits from one
Iwao uses a cloud computing the volumes of cylinders and brim to the other; it was completely
service to calculate  to more pyramids and also the area of a round. Its height was five cubits,
than 31 trillion decimal places. circle. The method used to find and a line of thirty cubits measured
the area of a circle was to find the its circumference.”
area of a square with sides that are In c. 250 bce, the Greek scholar
8 ⁄9 of the circle’s diameter. Using Archimedes developed an algorithm
this method implies that  is for determining the value of  based

Pi is the circumference of a circle

divided by its diameter. It is…

written as – the a constant – it has

16th letter of the a fixed value.
Greek alphabet. an irrational
number – it can’t a transcendental
be written as one number – it is
integer divided not an algebraic root.
by another.
 ANCIENT AND CLASSICAL PERIODS 63
See also: The Rhind papyrus 32–33 ■ Irrational numbers 44–45 ■ Euclid’s Elements 52–57 ■ Eratosthenes’ sieve 66–67
■ Zu Chongzhi 83 ■ Calculus 168–75 ■ Euler’s number 186–91 ■ Buffon’s needle experiment 202–03

on constructing regular polygons Although polygons had long been used to estimate the
that exactly fitted within (inscribed), circumference of circles, Archimedes was the first to use
or enclosed (circumscribed), a circle. inscribed (inside the circle) and circumscribed (outside the
circle) regular polygons to find upper and lower limits for .
He calculated upper and lower
limits for  by using Pythagoras’s
theorem – that the area of the
square of the hypoteneuse (the side
opposite the right angle) in a right-
angled triangle is equal to the sum
of the areas of the squares of the
other two sides – to establish the
relationship between the lengths
of the sides of regular polygons
when the number of sides was Pentagon Hexagon Octagon
doubled. This enabled him to
extend his algorithm to 96-sided same area as a given circle. Using
polygons. Determining the area of only a pair of compasses and a
a circle using a polygon with many straight edge, the Greeks would
sides had been proposed at least superimpose a square on a circle
200 years before Archimedes, but and then use their knowledge of
he was the first person to consider the area of a square to approximate The works of Archimedes are,
polygons that were both inscribed to the area of a circle. The Greeks without exception, works of
and circumscribed. were not successful with this mathematical exposition.
method, and in the 19th century Thomas L. Heath
Squaring the circle squaring the circle was proved to Historian and mathematician
Another method for estimating , be impossible, due to ’s irrational
“squaring the circle”, was a popular nature. This is why attempts to
challenge for mathematicians achieve an impossible task are
in ancient Greece. It involved sometimes known as “squaring
constructing a square with the the circle”. ❯❯

Archimedes Born in c. 287 bce in Syracuse, the same maximum radius

Sicily, the Greek polymath and height to be 3:2:1. Many
Archimedes excelled as a consider Archimedes to be a
mathematician and engineer, pioneer of calculus, which was
and is also remembered for his not developed until the 17th
“eureka” moment, when he century. He was killed by a
realized that the volume of water Roman soldier during the Siege
displaced by an object is equal to of Syracuse in 212 bce, despite
the volume of that object. Among orders that his life be spared.
his claimed inventions is the
Archimedes’ screw, a revolving Key works
screw-shaped blade in a cylinder,
which pushes water up a gradient. c. 250 bce On the Measurement
In mathematics, he used of a Circle
practical approaches to establish c. 225 bce On the Sphere and the
the ratio of the volumes of a Cylinder
cylinder, sphere, and cone with c. 225 bce On Spirals
64 CALCULATING PI
Another way mathematicians have During the 9th century, Arab
attempted to square the circle is to mathematician al-Khwarizmi
slice it into sections and rearrange used 31 ⁄ 7, 10, and 62,832 ⁄20,000 as
them into a rectangular shape (see values for , attributing the first
below). The area of the rectangle value to Greece and the other two
is r  1 ⁄2(2r)  r  r  r ² There is no end with pi. to India. English cleric Adelard
(where r is the radius of the circle I would love to try with of Bath translated al-Khwarizmi’s
and 2r is its diameter). The area more digits. work in the 12th century, renewing
of a circle is also r ². The smaller Emma Haruka Iwao an interest in the search for  in
the segments used, the closer the Japanese computer scientist Europe. In 1220, Leonardo of Pisa
shape is to a rectangle. (Fibonacci), who popularized
Hindu-Arabic numerals in his
The quest spreads book Liber Abaci (The Book of
More than 300 years after the death Calculation), 1202, computed 
of Archimedes, Ptolemy (c. 100– to be 864 ⁄275  3.141, a small
170 ce) determined  to be 3:8:30 improvement on Archimedes’s
(base-60), that is, 3  8 ⁄60  30 ⁄3,600 In India, the mathematician– approximation but not as accurate
 3.1416, which is just 0.007 per astronomer Aryabhata included as the calculations of Ptolemy, Zu
cent greater than the closest a method for obtaining  in his Chongzhi, or Aryabhata. Two
known value of . In China, 3 was Aryabhatiyam astronomical centuries later, Italian polymath
often used as the value of , until treatise of 499 ce: “Add 4 to 100, Leonardo da Vinci (1452–1519)
10 became common from the 2nd multiply by 8, and then add 62,000. proposed making a rectangle
century ce. The latter is 2.1 per cent By this rule the calculation of the whose length was the same as a
greater than . In the 3rd century, circumference of a circle circle’s circumference and whose
Wang Fau stated that a circle with with a diameter of 20,000 can height was half its radius to
a circumference of 142 had a be approached.” This works out as determine the area of the circle.
diameter of 45 – that is 142 ⁄45  3.15, [8(100  4)  62,000]  20,000  Archimedes’ method used in
just 1.4 per cent more than  – while 62,832  20,000  3.1416. ancient Greece for calculating 
Liu Hui used a 3,072-sided polygon Brahmagupta (c. 598–668 ce) was still being used in the late
to estimate  as 3.1416. In the 5th derived square root approximations 16th century. In 1579, French
century, Zu Chongzhi and his son of  using regular polygons with mathematician François Viète used
used a 24,576-sided polygon to 12, 24, 48, and 96 sides: 9.65, 393 regular polygons each with
calculate  as 355 ⁄113  3.14159292, 9.81, 9.86, and 9.87 respectively. 216 sides to calculate  to 10
a level of accuracy (to seven decimal Having established that 2  9.8696 decimal places. In 1593, Flemish
places) not achieved in Europe until to four decimal places, he simplified mathematician Adriaan van
the 16th century. these calculations to   10. Roomen (Romanus) used a polygon

r
r 2r

r
By arranging the segments of a circle in a
near-rectangular shape, it can be shown that the
area of a circle is r 2. The height of the “rectangle” is
approximately equal to the radius r of the circle, and the
width is half of the circumference (half of 2r, which is r).
 ANCIENT AND CLASSICAL PERIODS 65
Applying pi
Space scientists constantly
use  in their calculations.
For example, the length of
orbits at different altitudes
above a planet’s surface can
be worked out by using the
basic principle that if the
diameter of a circle is known,
its circumference can be
calculated by multiplying by .
In 2015, NASA scientists
applied this method to
compute the time it took the
spacecraft Dawn to orbit
Ceres, a dwarf planet in the
asteroid belt between Mars
and Jupiter.
When scientists at NASA's
Jet Propulsion Laboratory in
California wanted to know
The perimeter to height ratio of British mathematician William how much hydrogen might be
the Great Pyramid of Giza, in Egypt, is Rutherford computed 208 digits available beneath the surface
almost exactly , which might suggest of  using arctan series. of Europa, one of Jupiter's
that ancient Egyptian architects were The advent of calculators and
aware of the number. moons, they estimated the
electronic computers in the 20th hydrogen produced in a given
century made finding the digits unit area by first calculating
with 230 sides to compute  to of  much easier. In 1949, 2,037 Europa’s surface area, which
17 decimal places; three years digits of  were calculated in 70 is 4r2, as it is for any sphere.
later, German–Dutch professor of hours. Four years later, it took Since they knew Europa’s
mathematics Ludolph van Ceulen around 13 minutes to compute radius, calculating its surface
calculated  to 35 decimal places. 3,089 digits. In 1961, American area was easy.
The development of arctangent mathematicians Daniel Shanks It is also possible to work
series by Scottish astronomer– and John Wrench used arctan out the distance travelled
during one rotation of Earth
mathematician James Gregory series to compute 100,625 digits
by a person standing at a
in 1671, and independently by in under eight hours. In 1973, point on its surface using ,
Gottfried Leibniz in 1674, provided French mathematicians Jean providing the latitude of the
a new approach for finding . Guillaud and Martin Bouyer person’s position is known.
An arctangent (arctan) series is a achieved 1 million decimal
way of determining the angles in places, and in 1989, a billion
a triangle from knowledge of the decimal places were computed
length of its sides, and involves by Ukrainian–American brothers
radian measure, where a full turn David and Gregory Chudnovsky.
is 2 radians (equivalent to 360°). In 2016, Peter Trueb, a Swiss
Unfortunately, hundreds of particle physicist, used the
terms are needed to compute  to y-cruncher software to calculate
even a few decimal places using  to 22.4 trillion digits. A new
this series. Many mathematicians world record was set when
attempted to find more efficient computer scientist Emma Haruka Astrophysicists use  in their
calculations to determine the
methods to calculate  using Iwao calculated  to more than orbital paths and characteristics
arctan, including Leonhard Euler 31 trillion decimal places in of planetary bodies such as Saturn.
in the 18th century. Then, in 1841, March 2019. ■
66

WE SEPARATE
THE NUMBERS AS
IF BY SOME SIEVE
ERATOSTHENES’ SIEVE

I
n addition to calculating Prime numbers have exactly two
IN CONTEXT Earth’s circumference and the factors: 1 and the number itself.
distances from Earth to the The Greeks understood the
KEY FIGURE
Moon and Sun, the Greek polymath importance of primes as the
Eratosthenes (c. 276–c. 194 bce)
Eratosthenes devised a method building blocks of all positive
FIELD for finding prime numbers. Such integers. In his Elements, Euclid
Number theory numbers, divisible only by 1 stated many properties of both
and themselves, had intrigued composite numbers (integers
BEFORE mathematicians for centuries. above one that can be made by
c. 1500 bce The Babylonians By inventing his “sieve” to multiplying other integers) and
distinguish between prime eliminate non-primes – using a primes. These included the fact
and composite numbers. number grid and crossing off that every integer can be written
multiples of 2, 3, 5, and above – as a product of prime numbers or
c. 300 bce In Elements (Book
Eratosthenes made prime numbers is itself a prime. A few decades
IX proposition 20), Euclid considerably more accessible. later, Eratosthenes developed his
proves that there are infinitely
many prime numbers.
AFTER
Early 19th century Carl Eratosthenes developed his “sieve” as a method to
speed up the process of finding prime numbers.
Friedrich Gauss and French
mathematician Adrien-Marie
Legendre independently
produce a conjecture about
the density of primes. Numbers are written out in a table.
1859 Bernhard Riemann
states a hypothesis about the
distribution of prime numbers.
The hypothesis has been used
The method leaves you Multiples of prime
to prove many other theories with a grid where prime numbers are
about prime numbers, but it numbers are clearly systematically
has not yet been proved. identified. crossed out.
 ANCIENT AND CLASSICAL PERIODS 67
See also: Mersenne primes 124 ■ The Riemann hypothesis 250–51
■ The prime number theorem 260–61 ■ Finite simple groups 318–19

Eratosthenes’ method starts with a table of consecutive numbers. First, 1 is

crossed out. Then all multiples of 2 are crossed out except 2 itself. The same is
then done for multiples of 3, 5 and 7. Multiples of any number higher than 7 are
already crossed out, since 8, 9, and 10 are composites of 2, 3, and 5.
Prime
numbers 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
1 and
composite 21 22 23 24 25 26 27 28 29 30
numbers
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
Eratosthenes
51 52 53 54 55 56 57 58 59 60
Born around 276 bce in
61 62 63 64 65 66 67 68 69 70 Cyrene, a Greek city in
71 72 73 74 75 76 77 78 79 80 Libya, Eratosthenes studied
in Athens and became a
81 82 83 84 85 86 87 88 89 90 mathematician, astronomer,
91 92 93 94 95 96 97 98 99 100 geographer, music theorist,
literary critic, and poet. He
was the chief librarian at the
method, which can be extended to have already been removed, and Library of Alexandria, the
greatest academic institution
uncover all primes. Using a number so on for all successive numbers.
of the ancient world. He
grid for 1 to 100 (see above), it is There are only 25 prime numbers
is known as the father of
clear that 1 is not a prime number up to 100 – starting with 2, 3, 5, 7, geography for founding
as its only factor is 1. The first and 11, and ending with 97 – all and naming the subject
prime number – and also the only identified by simply removing every as an academic discipline
even prime – is 2. As all other even multiple of 2, 3, 5, and 7. and developing much of
numbers are divisible by 2, they the geographical language
cannot be primes, so all other The search continues used today.
primes must be odd. The next Prime numbers attracted the Eratosthenes also
prime, 3, has only two factors, so attention of mathematicians from recognized that Earth is a
all the other multiples of 3 cannot the 17th century onwards, when sphere and calculated its
be primes. The number 4 (2  2) figures such as Pierre de Fermat, circumference by comparing
has already had its multiples Marin Mersenne, Leonhard Euler, the angles of elevation of the
removed, since they are all even. and Carl Friedrich Gauss probed Sun at noon at Aswan in
The next prime is 5, so all other further into their properties. southern Egypt and at
Alexandria in the north of
multiples of 5 cannot be prime. Even in the age of computers,
the country. In addition, he
The number 6 and all its multiples determining whether a large produced the first world map
have been removed from the list of number is prime remains that featured meridian lines,
potential primes, as they are even highly challenging. Public key the Equator, and even polar
multiples of 3. The next prime is cryptography – the use of two large zones. He died around 194 bce.
7, and removing its multiples primes to encrypt a message – is
eliminates 49, 77, and 91. All the the basis of all internet security.
Key works
multiples of 9 have gone, as they If hackers ever do figure out a
are multiples of 3, and all the simple way of determining the Mensuram orae ad terram (On
multiples of 10 have been removed, prime factorization of very large the Measurement of the Earth)
because they are the even multiples numbers, a new system will need Geographika (Geography)
of 5. The multiples of 11 up to 100 to be devised. ■
68

A GEOMETRICAL
TOUR DE FORCE
CONIC SECTIONS

O
f the many pioneering intact. It was written in eight
IN CONTEXT mathematicians produced volumes, of which seven survive:
by ancient Greece, books 1–4 in Greek, and books 5–7
KEY FIGURE
Apollonius of Perga was one of the in Arabic. The work was designed
Apollonius of Perga
most brilliant. He began studying to be read by mathematicians
(c. 262–190 bce)
mathematics after Euclid’s great already well versed in geometry.
FIELD work Elements had emerged and he
Geometry employed the Euclidian method of A new geometry
taking “axioms” – statements taken Early Greek mathematicians such
BEFORE to be true – as starting points for as Euclid concentrated on the line
c. 300 bce Euclid’s 13-volume further reasoning and proofs. and the circle as the purest
Elements sets out the Apollonius wrote on many geometric forms. Apollonius viewed
propositions that form the subjects, including optics (how these in three-dimensional terms:
basis of plane geometry. light rays travel) and astronomy, as if a circle is combined with all lines
well as geometry. Much of his work that emanate from it, above or
c. 250 bce In On Conoids and survives only in fragments, but his below its plane, and those lines
Spheroids, Archimedes deals most influential, Conics, is relatively pass through the same fixed point –
with the solids created by the the vertex – a cone is created. By
revolution of conic sections slicing that cone in different ways,
about their axes. a series of curves, known as conic
AFTER sections, can be produced.
c. 1079 ce Persian polymath In Conics, Apollonius expounded
in minute detail this new world of
Omar Khayyam uses I have sent my son… to bring geometric construction, studying
intersecting conics to solve you… the second book of my and defining the properties of conic
algebraic equations. Conics. Read it carefully and sections. He based his workings on
1639 In France, 16-year-old communicate it to such others the assumption of two cones joined
Blaise Pascal asserts that as are worthy of it. at the same vertex, with the area
where a hexagon is inscribed Apollonius of Perga of their circular bases potentially
in a circle, the opposite sides stretching to infinity. To three of
of the hexagon meet at three the conic sections he gave the
points on a straight line. names ellipse, parabola, and
hyperbola. An ellipse occurs when
a plane intersects a cone on a slant.
 ANCIENT AND CLASSICAL PERIODS 69
See also: Euclid’s Elements 52–57 ■ Coordinates 144–51 ■ The area under a cycloid 152–53 ■ Projective geometry 154–55
■ The complex plane 214–15 ■ Non-Euclidean geometries 228–29 ■ Proving Fermat’s last theorem 320–23

A parabola emerges if the cut is Axis

parallel to the edge of the cone, and
a hyperbola results when the plane is
Plane
vertical. Although he saw the circle
as one of the four conic sections, it
is really an ellipse with the plane Apex
perpendicular to the axis of the cone.
CIRCLE ELLIPSE PARABOLA
Paving the way for others
In his description of these four
geometric objects, Apollonius
used no algebraic formulae and
no numbers. However, his view of
a conic curve as a set of ordered
parallel lines emanating from
an axis looked towards the later
creation of coordinate system
geometry. He did not achieve the HYPERBOLA SINGLE POINT PAIR OF STRAIGHT LINES
kind of precision that would come
1,800 years later with the work of When a plane intersects a cone, it creates a conic section.
As well as the sections described by Apollonius, this can be a
French mathematicians René
single point, where the plane cuts across the apex (top vertex),
Descartes and Pierre de Fermat, or straight lines cutting through the apex at an angle.
but he did get close to coordinate
representations of his conic curves.
Some things held Apollonius back: quadrants – with both positive and the Middle Ages. His work was
he did not use negative numbers, negative coordinates – Apollonius then rediscovered in Europe
nor did he explicitly work with effectively worked in just one. during the Renaissance, leading
zero. So while the two-dimensional Apollonius’s studies inspired mathematicians to develop the
Cartesian geometry developed many of the advances in geometry analytic geometry that helped
by Descartes worked across four seen in the Islamic world during to fuel the scientific revolution. ■

Apollonius of Perga known to Euclid, but the later

works were a significant
Little is known about the life advance in geometry.
of Apollonius. He was born in Beyond his work with conic
c.262 bce in Perga, a centre for sections, Apollonius is credited
[Conic sections are] the worship of the goddess with estimating the value of
the necessary key with Artemis, in southern Anatolia pi more accurately than his
which to attain the knowledge (now part of Turkey). After contemporary Archimedes,
of the most important crossing the Mediterranean and with being the first to state
laws of nature. to Egypt, he was taught by that a spherical mirror does not
Alfred North Whitehead Euclidean scholars in the great focus the sun’s rays, while a
British mathematician cultural city of Alexandria. parabolic mirror does.
It is thought that all eight
volumes of Conics were
compiled while Apollonius Key work
was in Egypt. The first volumes
produced little that was not c. 200 bce Conics
THE ART
OF MEASURING
TRIANGLES
TRIGONOMETRY
72 TRIGONOMETRY

IN CONTEXT
Trigonometry is the study of the relationship between
KEY FIGURE the sides and angles of triangles.
Hipparchus (c. 190–120 bce)
FIELD
Geometry
BEFORE If two angles are known,
The three angles in any
c. 1800 bce The Babylonian triangle add up to 180. the third angle can be
Plimpton 322 tablet contains determined.
a list of Pythagorean triples,
long before Pythagoras devised
his formula a2  b2  c2.
c. 1650 bce The Egyptian
The ratios of the sides of a right-angled triangle are
Rhind papyrus includes a
called trigonometric ratios.
method for calculating the
slope of a pyramid.
6th century bce In ancient
Greece, Pythagoras discovers
his theorem relating to the If the length of one side of a triangle is known and its
geometry of triangles. angles are known, the length of the other sides
can be determined.
AFTER
500 ce In India, the first
trigonometric tables are used.
all right-angled triangles contain alongside another set of numbers
1000 ce In the Islamic world, two shorter sides (which may or that resemble the ratios of the
mathematicians are using all may not be of equal length) and a squares of sides. The tablet’s
the various ratios between the diagonal, or hypotenuse, which is original purpose is unknown, but it
sides and angles of triangles. longer than either of the others; may have been used as a practical
all triangles contain three angles; manual for measuring dimensions.
and right-angled triangles have

T
rigonometry, a term based one angle of 90.
on the Greek words for
“triangle” and “measure”, The Plimpton tablet
is of immense importance in both In the early 1900s, an examination
the historical development of of triangles, dating back to around
mathematics and in the modern 1800 bce, was discovered on an Even if he did not invent it,
world. Trigonometry is one of the ancient Babylonian clay tablet. Hipparchus is the first person
most useful of all the mathematical The tablet, bought by American of whose systematic use of
disciplines, enabling people to publisher George Plimpton in 1923 trigonometry we have
navigate the world, to understand and known as Plimpton 322, is documentary evidence.
electricity, and to measure the etched with numerical information Sir Thomas Heath
height of mountains. relating to right-angled triangles. Its British historian of mathematics
Since antiquity, civilizations exact significance is debated, but
have appreciated the need for right the information appears to include
angles in architecture. This led Pythagorean triples (three positive
mathematicians to analyse the numbers representing the lengths
properties of right-angled triangles: of sides of a right-angled triangle),
 ANCIENT AND CLASSICAL PERIODS 73
See also: The Rhind papyrus 32–33 ■ Pythagoras 36–43 ■ Euclid’s Elements 52–57 ■ Imaginary and complex numbers
128–31 ■ Logarithms 138–41 ■ Pascal’s triangle 156–61 ■ Viviani’s triangle theorem 166 ■ Fourier analysis 216–17

At around the same time as the generally regarded as the founder

ancient Babylonians, Egypt’s of trigonometry, was particularly
mathematicians were developing interested in triangles inscribed
an interest in geometry. This was within circles and spheres, and
driven not just by their monumental the relationship between angles
building programme but also by the and lengths of chords (straight
annual flooding of the River Nile, lines drawn between two points
which required them to mark out on a circle – or on any curve).
the areas of fields each time the Hipparchus compiled what was
floods subsided. Egyptian interest effectively the first true table of
is evident in the Rhind papyrus, trigonometric values.
a scroll that contains a set of tables
relating to fractions. One of these Ptolemy’s contribution
tables poses the question: “If a Around 300 years later, in the
pyramid is 250 cubits high and the Egyptian city of Alexandria,
side of its base is 360 cubits long, the gifted Greco-Roman polymath
In the medieval period, astrolabes
what is its seked?” The word seked Claudius Ptolemaeus, better known applied trigonometric principles to
means slope, so the problem is as Ptolemy, wrote a mathematical measure the position of celestial
purely trigonometrical. treatise called the Syntaxis bodies. Hipparchus is credited with
Mathematikos (later renamed the inventing the device.
Hipparchus sets out rules Almagest by Islamic scholars).
Influenced by Babylonian theories In this work, Ptolemy further
on angles, the ancient Greeks developed the ideas of Hipparchus used the Babylonian system of
developed trigonometry as a branch on triangles and chords of circles, numbers known as the sexagesimal
of mathematics that was governed building formulae that would allow system, based on the number 60.
by definite rules rather than the the prediction of the position of the Ptolemy’s work was developed
tables of numbers relied on by the Sun and other “heavenly bodies” further in India, where the growing
earliest mathematicians. In the based on the assumption of circular discipline of trigonometry was
2nd century bce, the astronomer orbits around Earth. Ptolemy, like regarded as part of astronomy. The
and mathematician Hipparchus, the mathematicians before him, Indian mathematician Aryabhata ❯❯

Hipparchus Hipparchus was born in Nicaea Hipparchus’s most notable

(now Iznik in Turkey) in 190 bce. contribution to astronomy was
Although little is known of his his work Sizes and Distances
life, he achieved fame as an (now lost, but used by Ptolemy),
astronomer from the studies he on the orbits of the Sun and
carried out while living on the Moon, which enabled him to
island of Rhodes. His findings calculate the dates of the
were immortalized in Ptolemy’s equinoxes and solstices. He
Almagest, where he is described also compiled a star catalogue,
as “a lover of truth”. which may be the one used
The only work of Hipparchus by Ptolemy in Almagest.
to survive was his commentary Hipparchus died in 120 bce.
on the Phaenomena of the poet
Aratus and the mathematician Key work
and astronomer Eudoxus,
criticizing the inaccuracy of their 2nd century bce Sizes and
descriptions of constellations. Distances
74 TRIGONOMETRY
not just for astronomy, but also for
Types of trigonometry religious purposes, since it was
important that Muslims knew the
a = opposite position of the holy city of Mecca
b = adjacent from anywhere in the world.
c = hypotenuse In the 12th century ce, Indian
a
mathematician and astronomer
Bhaskara II invented the study
c b c of spherical trigonometry. This
a explores triangles and other
shapes on the surface of a sphere
rather than on a plane.
In later centuries, trigonometry
b became invaluable in navigation
as well as astronomy. Bhaskara II’s
Planar trigonometry is the study Spherical trigonometry is the study work, along with the ideas in
of triangles on a plane (a flat, 2-D of triangles on a sphere (a curved, 3-D Ptolemy’s Almagest, were valued
surface). Planar trigonometry is used surface). It is used by astronomers to by the Islamic scholars of the
by architects, for example, to ensure calculate the positions of celestial
buildings are stable, and by physicists bodies, and in navigation to calculate medieval world, who had begun
to model motion. latitude and longitude. studying trigonometry well
before Bhaskara II.

(474–550 ce) pursued the study of his own contributions to geometry Aid to astronomy
chords to produce the first table and trigonometry, including what Together with the developments in
of what is now known as the sine is now known as Brahmagupta’s trigonometry, there was a gradual
function (all the possible values of formula. This is used to find the and corresponding shift in the way
sine/cosine ratios for determining area of cyclic quadrilaterals, which people viewed the heavens. From
the unknown length of the side of are four-sided shapes inscribed passively observing and recording
a triangle when the lengths of the within a circle. This area can also the patterns in the movement
hypotenuse – the triangle’s longest be found with a trigonometric of celestial bodies, scholars
side – and the side opposite the method if the quadrilateral is split began to model that movement
angle are known). into two triangles. mathematically so that they could
In the 7th century ce, another predict future astronomical events
great Indian mathematician and Islamic trigonometry with ever greater accuracy. The
astronomer, Brahmagupta, made Brahmagupta had already created
a table of sine values, but in the
9th century ce, Persian astronomer
and mathematician Habash
al-Hasib (“Habash the Calculator”)
produced some of the first sine,
cosine, and tangent tables to A logarithmic table is a small
Trigonometry, like other table by the use of which we
branches of mathematics, calculate the angles and sides of
triangles. Around the same time, can obtain knowledge of all
was not the work of any one geometrical dimensions
al-Battani (Albatenius) developed
man, or nation. and motions in space.
Ptolemy’s work on the sine function
Carl Benjamin Boyer and applied it to astronomical John Napier
American historian of mathematics
calculations. He recorded highly
accurate observations of celestial
objects from Raqqah, Syria. The
motivation among Arab scholars
for developing trigonometry was
 ANCIENT AND CLASSICAL PERIODS 75
study of trigonometry purely as an the new system of imaginary
aid to astronomy persisted well numbers that had been invented
into the 16th century, when new by Italian mathematician Rafael
developments in Europe began to Bombelli in 1572.
gain momentum. De Triangulis At the end of the 16th century,
Omnimodis (On Triangles of all Italian physicist and astronomer
Kinds) was published in 1533. Galileo Galilei used trigonometry to
Written by German mathematician model the trajectories of projectiles
Johannes Müller von Königsberg, on which gravity was acting. The
known as Regiomontanus, it was a same equations are still used to
compendium of all known theorems project the motion of rockets and
for finding sides and angles of both missiles into the atmosphere today.
planar (2-D) and spherical triangles Also in the 16th century, Dutch
(those formed on the surface of a cartographer and mathematician
3-D sphere). The publication of this Gemma Frisius used trigonometry
A network of triangulation
work marked a turning point for to determine distances, thus stations such as this stone “trig
trigonometry. It was no longer enabling accurate maps to be point” in Wales was launched by the
merely a branch of astronomy but created for the first time. Ordnance Survey in 1936 to accurately
a key component of geometry. map the island of Great Britain.
Trigonometry was to develop New developments
further still; although geometry Developments in trigonometry
was its natural home, it was also gathered pace in the 17th century. went a step further than Vieté
increasingly applied to solve Scottish mathematician John and showed how trigonometric
algebraic equations. French Napier’s discovery of logarithms functions could be used in the
mathematician François Viète in 1614 enabled the compilation of analysis of complex numbers.
showed how algebraic equations accurate sine, cosine, and tangent The latter comprised a real part
could be solved using trigonometric tables. In 1722, Abraham de and an imaginary part, and were
functions, in conjunction with Moivre, a French mathematician, to be of great significance in the
development of mechanical and
electrical engineering. Leonhard
opposite opposite Euler used de Moivre’s findings to
sine tangent
sin  = tan  = derive the “most elegant equation
formula hypotenuse formula adjacent in mathematics”: ei  + 1 = 0, also
known as Euler’s identity.
cosine adjacent In the 18th century, Joseph
formula cos  =
hypotenuse Fourier applied trigonometry to
his research into different forms of
E
TE NUS waves and vibrations. The “Fourier
OPPOSITE

H YPO trigonometry series” has been used

widely in scientific fields such as
optics, electromagnetism, and,
more recently, quantum mechanics.
From its early beginnings, when
 90°
the Babylonians and ancient
Egyptians pondered the lengths
ADJACENT
of shadows cast by a stick in the
ground, through architecture and
To find the unknown angle () in a right-angled triangle, the sine formula
is used when the lengths of the opposite (opposite ) and the hypotenuse
astronomy to modern applications,
are known; the cosine formula is used when the lengths of the adjacent and trigonometry has formed a part of
hypotenuse are known; and the tangent formula is used when the lengths the language of mathematics in
of the opposite and adjacent are known. modelling the Universe. ■
76
IN CONTEXT

NUMBERS
KEY CIVILIZATION
Ancient Chinese
(c. 1700 bce–c. 600 ce)

CAN BE
FIELD
Number systems
BEFORE
c. 1000 bce In China, bamboo

LESS THAN
rods are first used to denote
numbers, including negatives.
AFTER
628 ce The Indian

NOTHING
mathematician Brahmagupta
provides rules for arithmetic
with negative numbers.
1631 In Practice of the Art of
Analysis, published 10 years

NEGATIVE NUMBERS after his death, British

mathematician Thomas
Harriott accepts negative
numbers in algebraic notation.

W
hile practical notions
of negative quantities
were used from ancient
times, particularly in China,
negative numbers took far longer to
be accepted within mathematics.
Ancient Greek thinkers and many
later European mathematicians
regarded negative numbers – and
the concept of something being
less than nothing – as absurd. Only
in the 17th century did European
mathematicians begin to fully
accept negative numbers.

Chinese rod system

The earliest ideas of negative
quantities seem to have arisen in
commercial accounting: the seller
received money for what had been
sold (a positive quantity), and the
buyer spent the same amount,
 ANCIENT AND CLASSICAL PERIODS 77
See also: Positional numbers 22–27 ■ Diophantine equations 80–81
■ Zero 88–91 ■ Algebra 92–99 ■ Imaginary and complex numbers 128–31
Mathematics in
ancient China

In the Chinese rod numeral system, red indicates positive numbers, Jiuzhang suanshu, or The Nine
while black indicates negative numbers. To make the number being Chapters on the Mathematical
represented as clear as possible, horizontal and vertical symbols are used Art, reveals the mathematical
alternately – for example, the number 752 would use a vertical 7, then a methods known to the ancient
horizontal 5, followed by a vertical 2. Blank spaces represent zero. Chinese. It is written as a
collection of 246 practical
problems and their solutions.
Positive 0 1 2 3 4 5 6 7 8 9
The first five chapters are
mostly about geometry (areas,
Vertical lengths, and volumes) and
arithmetic (ratios, and square
and cube roots). Chapter six
Horizontal covers taxes, and includes the
ideas of direct, inverse, and
compound proportions, most
of which did not appear in
Negative 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 Europe until around the 16th
century. Chapters seven and
Vertical
eight deal with solutions to
linear equations, including the
rule of “double false position”,
whereby two test (or “false”)
Horizontal values for the solution to a
linear equation are used in
repeated steps to yield the
actual solution. The final
resulting in a deficit (a negative When rods of different colours were chapter concerns applications
quantity). For their commercial added together, they cancelled of the “Gougu” (equivalent to
arithmetic, the ancient Chinese each other out – like income Pythagoras’s theorem), and the
used small bamboo rods, laid erasing a debt. The polarized solving of quadratic equations.
out on a large board. Positive nature of positive numbers (red
and negative quantities were rods) and negative numbers
represented by rods of different (black rods) was also in tune with
colours and could be added the Chinese concept that opposing
together. The Chinese military but complementary forces – yin and
strategist Sun Tzu, who lived yang – governed the Universe.
around 500 bce, used such rods to
make calculations before battles. Fluctuating fortunes
By 150 bce, the rod system Over a period of several centuries,
had developed into alternating starting around 200 bce, the ancient
horizontal and vertical rods in sets Chinese produced a book of
of up to five. Later, during the Sui collected scholarship called The
dynasty (581–618 ce), the Chinese Nine Chapters on the Mathematical
also used triangular rods for Art (see box). This work, which
positive quantities and rectangular encapsulated the essence of their
rods for negative quantities. The mathematical knowledge, included
Temperature readings on the
system was employed for trading algorithms that assumed negative Celsius scale display negative numbers
and tax calculations: amounts quantities were possible – for to show when something such as an
received were represented by example, as solutions to problems ice crystal is colder than 0C – the
red rods, and debts by black rods. on profit and loss. ❯❯ point at which water freezes.
78 NEGATIVE NUMBERS
The Persian mathematican and
 -4 -3 -2 -1 0 1 2 3 4 poet al-Khwarizmi (c. 780–c. 850) –
whose theories, particularly on
-4 16 12 8 4 0 -4 -8 -12 -16 algebra, influenced later European
mathematicians – was familiar
-3 12 9 6 3 0 -3 -6 -9 -12
with the rules of Brahmagupta
-2 8 6 4 2 0 -2 -4 -6 -8 and understood the use of negative
numbers for dealing with debts.
-1 4 3 2 1 0 -1 -2 -3 -4 However, he could not accept the
use of negative numbers in algebra,
0 0 0 0 0 0 0 0 0 0 believing them to be meaningless.
Instead, al-Khwarizmi followed
1 -4 -3 -2 -1 0 1 2 3 4 geometric methods to solve linear
or quadratic equations.
2 -8 -6 -4 -2 0 2 4 6 8
Accepting the negative
3 -12 -9 -6 -3 0 3 6 9 12 Throughout the Middle Ages,
European mathematicians
4 -16 -12 -8 -4 0 4 8 12 16 remained unsure of negative
quantities as numbers. This was
A negative multiplied by a negative makes a positive. Positive number still the case in 1545 when Italian
This is why all positive numbers have two square roots Negative number polymath Gerolamo Cardano
(a positive and a negative) and negative numbers have no
published his Ars Magna (The
real square roots – because a positive number squared is
positive, and a negative number squared is also positive. Great Art), in which he explained
how to solve linear, quadratic, and
cubic equations. He could not
In contrast, the mathematics even used a symbol to indicate exclude negative solutions to his
of ancient Greece was based negative numbers. Like the ancient equations and even used a sign,
on geometry and geometrical Chinese, Brahmagupta looked at “m”, to denote a negative number.
magnitudes, or their ratios. As numbers in financial terms, as He could not, however, accept the
these quantities – actual lengths, “fortunes” (positive) and “debts” value of negative numbers, calling
areas, and volumes – can only be (negative), and stated the following them “fictitious”. René Descartes
positive, the idea of a negative rules for multiplying with positive (1596–1650) also accepted negative
number did not make sense to and negative quantities: quantities as solutions to equations
Greek mathematicians. The product of two fortunes is but referred to them as “false roots”
By the time of Diophantus, a fortune. The product of two rather than true numbers.
around 250 ce, linear and quadratic debts is a fortune. The product
equations were used to solve of a debt and a fortune is a debt.
problems, but any unknown The product of a fortune and
quantity was still represented a debt is a debt.
geometrically – by a length. So It makes no sense to find the
the idea of negative numbers as product of two piles of coins, as Negative numbers are
solutions to these equations was only the actual quantities can be evidence of inconsistency
still seen as an absurdity. multiplied, not the money itself (just or absurdity.
An important advance in as you cannot multiply apples by
Augustus De Morgan
the arithmetical use of negative apples). Brahmagupta was therefore British mathematician
numbers came around 400 years performing arithmetic with positive
later from India, in the work of and negative quantities, while
the mathematician Brahmagupta using fortunes and debts as a way
(c. 598–668). He set out arithmetic to try to understand what negative
rules for negative quantities, and numbers represented.
 ANCIENT AND CLASSICAL PERIODS 79
English mathematician John Wallis
(1616–1703) gave some meaning to
negative numbers by extending the
In 15th-century
number line below zero. This way Europe, the letters p The + and - signs
of seeing numbers as points on a and m are used for plus are introduced in the
line finally led to the acceptance and minus. 16th century.
of negative numbers on equal
terms with positive numbers, and
by the end of the 19th century
they had been formally defined
within mathematics, separate
from notions of quantities. Today,
negative numbers are used in
many areas, ranging from banking But negative numbers are seen as absurd
and viewed with hostility and suspicion.
and temperature scales to the
charge on subatomic particles.
Any ambiguity about their status
in mathematics is long gone. ■

Investors rush to withdraw their

money from the Seamen’s Savings It is not until the 17th century that negative numbers
Bank in New York in 1857. The panic are accepted in Europe, when they are placed on a
was caused by American banks number line for the first time.
loaning out many millions of dollars (a
negative quantity) without the reserves
(a positive quantity) to back this up.
80

THE VERY
FLOWER OF
ARITHMETIC
DIOPHANTINE EQUATIONS

IN CONTEXT Diophantus tried to solve

KEY FIGURE equations with more than two unknown
Diophantus (c. 200–c. 284 ce) quantities and only integer
or rational number solutions.
FIELD
Algebra
BEFORE While some have a simple Equations like this are
c. 800 bce The Indian scholar solution, most have many now known as
Baudhayana finds solutions to solutions – or none at all. Diophantine equations.
some “Diophantine” equations.
AFTER
c. 1600 François Viète lays
the foundations for solutions
of Diophantine equations. Diophantine equations have proved endlessly
fascinating for mathematicians.
1657 Pierre de Fermat writes
his last theorem (about a
Diophantine equation) in

I
his copy of Arithmetica.
n the 3rd century ce, the Greek as Diophantine equations. Today,
1900 The 10th problem on mathematician Diophantus, a the equations are considered to be
David Hilbert’s list of unsolved pioneer of number theory and one of the most interesting areas
research problems is the quest arithmetic, created a prodigious of number theory.
to find an algorithm to solve all work called Arithmetica. In 13 Diophantine equations are a
Diophantine equations. volumes, only six of which have type of polynomial – an equation in
survived, he explored 130 problems which the powers of the variables
1970 Mathematicians in involving equations and was the (unknown quantities) are integers,
Russia show that there is no first person to use a symbol for an such as x3 + y4 = z5. The aim of
algorithm that can solve all unknown quantity – a cornerstone Diophantine equations is to find all
Diophantine equations. of algebra. It is only in the past 100 the variables, but solutions must be
years that mathematicians have integers or rational numbers (those
fully explored what are now known that can be written as one integer
 ANCIENT AND CLASSICAL PERIODS 81
See also: The Rhind papyrus 32–33 ■ Pythagoras 36–43 ■ Hypatia 82 ■ The equals sign and other symbology 126–27
■ 23 problems for the 20th century 266–67 ■ The Turing machine 284–89 ■ Proving Fermat’s last theorem 320–23

The Arithmetica of Diophantus

strongly influenced 17th-century
mathematicians as the study of modern
algebra developed. This volume of the
book was published in Latin in 1621.

The symbolism that

Diophantus introduced asserted in a marginal note in 1657
for the first time… provided and British mathematician Andrew
a short and readily Wiles finally proved in 1994.
comprehensible means of
expressing an equation. A source of fascination
Diophantine equations are vast in
Kurt Vogel number and form, and mostly very
German mathematical historian
difficult to solve. In 1900, David
Hilbert suggested that the question
of whether or not they could all
be solved was one of the greatest
challenges facing mathematicians.
The equations are now grouped
in three classes: those with no
divided by another, such as 8 ⁄3). solution, those with a finite number
In Diophantine equations, the of solutions, and those with an
coefficients – integers such as the infinite number of solutions. Rather had studied for years, concluding
4 in 4x, that multiply a variable – are than finding solutions, however, that no general algorithm to solve
also rational numbers. Diophantus mathematicians are often more a Diophantine equation exists. Yet
only used positive numbers, but interested in discovering whether studies continue, as the fascination
mathematicians now look for solutions exist at all. In 1970, of these equations is largely
negative solutions as well. Russian mathematician Yuri theoretical. Mathematicians, who
Matiyasevich settled Hilbert’s are driven by curiosity, believe
The quest for solutions query, which he and three others there is still more to discover. ■
Many of the problems now called
Diophantine equations were known Diophantus around 500 ce, contains one
well before Diophantus’s time. In number problem purporting to
India, mathematicians explored Little is known about the life be an epitaph to Diophantus
some of them from around 800 bce, of the Greek mathematician that appeared on his tombstone.
as the ancient Shulba Sutras texts and philosopher Diophantus, Written as a puzzle, it suggests
reveal. In the 6th century bce, but he was probably born in he married at the age of 35, and
Pythagoras created his quadratic Alexandria, Egypt, in c. 200 ce. five years later had a son, who
equation for calculating the sides of His 13-volume Arithmetica was died at the age of 40 when he
a right-angled triangle; its x2 + y2 = well-received – the Alexandrian was half his father’s age.
z2 form is a Diophantine equation. mathematician Hypatia wrote Diophantus is then said to have
Diophantine equations of the about the first six volumes – lived a further four years, dying
but fell into relative obscurity at the age of 84.
kind x + y = z may look simple
n n n
until the 16th century, when
to calculate, but only those with
interest in his ideas was revived.
squares are solvable. If the power The Greek Anthology, a Key work
(n in the equation) is greater than compilation of mathematical
2, the equation has no integer games and verses published c. 250 ce Arithmetica
solutions for x, y, and z – as Fermat
82

AN INCOMPARABLE
STAR IN THE
FIRMAMENT OF
WISDOM
HYPATIA

H
istory mentions only a
IN CONTEXT few pioneering female
mathematicians in the
KEY FIGURE
ancient world, among them Hypatia
Hypatia of Alexandria
of Alexandria. An inspirational
(c. 355–415 ce)
teacher, she was appointed head of
FIELDS the city’s Platonist school in 400 ce.
Arithmetic, geometry Hypatia is not known to have
contributed any original research,
BEFORE but she is credited with editing and
6th century bce Pythagoras’s writing commentaries on several
wife Theano and other women classic mathematical, astronomical,
actively participate in the and philosophical texts. It is likely
The Alexandrian scholar Hypatia,
Pythagorean community. that she helped her father, Theon, depicted here in an 1889 painting by
a respected Alexandrian scholar, Julius Kronberg, was revered as a
c. 100 bce Mathematician and to produce his definitive edition heroic martyr after her murder. She
astronomer Aglaonike of of Euclid’s Elements, and his later became a symbol for feminists.
Thessaly wins renown for her Almagest and Handy Tables of
ability to predict lunar eclipses. Ptolemy. She also continued his
AFTER project of preserving and expanding Hypatia won great renown for her
1748 Italian mathematician the classic texts, in particular teaching, scientific knowledge, and
Maria Agnesi writes the first providing commentaries on wisdom, but in 415 she was killed
Diophantus’s 13-volume Arithmetica, by Christian zealots for her “pagan”
textbook to explain differential
and Apollonius’s work on conic philosophy. As attitudes towards
and integral calculus.
sections. Hypatia may have women in academia became less
1874 Russian mathematician intended these editions to serve tolerant, mathematics and astronomy
Sofia Kovalevskaya is the first as textbooks for students, as she would be almost exclusively male
woman to be awarded a offered commentaries providing preserves until the Enlightenment
doctorate in mathematics. clarification, and developed some opened up new opportunities for
of the concepts further. women in the 18th century. ■
2014 Iranian mathematician
Maryam Mirzakhani is the first See also: Euclid’s Elements 52–57 ■ Conic sections 68–69 ■ Diophantine
woman to win the Fields Medal. equations 80–81 ■ Emmy Noether and abstract algebra 280–81
 ANCIENT AND CLASSICAL PERIODS 83

THE CLOSEST
APPROXIMATION OF
PI FOR
ZU CHONGZHI
A MILLENNIUM

L
ike their counterparts in an even more accurate value for .
IN CONTEXT Greece, mathematicians in Using a 12,288-sided polygon,
ancient China realized the he calculated that  is between
KEY FIGURE
importance of  (pi) – the ratio of 3.1415926 and 3.1415927, and
Zu Chongzhi (429–501 ce)
a circle’s circumference to its suggested two fractions to express
FIELD diameter – in geometric and other the ratio: the Yuelü, or approximate
Geometry calculations. Various values for  ratio, of 22 ⁄ 7, which had been in
were suggested from the 1st use for some time; and his own
BEFORE century ce onwards. Some were calculation, the Milü, or close ratio,
c. 1650 bce The area of a circle sufficiently accurate for practical of 355 ⁄113. This later became known
is calculated using  as (16 ⁄9)2 ≈ purposes, but several Chinese as “Zu’s ratio”. Zu’s calculations of
3.1605 in the Rhind papyrus. mathematicians sought more  were not bettered until European
precise methods for determining . mathematicians set about the task
c. 250 bce Archimedes finds an
In the 3rd century, Liu Hui during the Renaissance, almost a
approximate value for  using
approached the task using the millennium later. ■
a polygon algorithm method. same method as Archimedes –
AFTER drawing regular polygons with
c. 1500 Indian astronomer increasing numbers of sides inside
Nilakantha Somayaji uses and outside a circle. He found that
an infinite series (the sum of a 96-sided polygon allowed a
terms of an infinite sequence, calculation of  as 3.14, but by I cannot help thinking
such as ½ + ¼ + 1 ⁄8 + 1 ⁄16) to repeatedly doubling the number that Zu Chongzhi was a
of sides up to 3,072, he reached genius of Antiquity.
compute .
a value of 3.1416.
Takebe Katahiro
1665–66 Isaac Newton Japanese mathematician
calculates  to 15 digits. More precision
In the 5th century, astronomer and
1975–76 Iterative algorithms mathematician Zu Chongzhi, who
allow computer calculations of was renowned for his meticulous
 to millions of digits. calculations, set about obtaining

See also: The Rhind papyrus 32–33 ■ Irrational numbers 44–45 ■ Calculating
pi 60–65 ■ Euler’s identity 197 ■ Buffon’s needle experiment 202–03
THE MID
AGES
500–1500
DLE
86 INTRODUCTION

The Indian mathematician The House of Wisdom is

Brahmagupta establishes established in Baghdad, Al-Khwarizmi and al-Kindi
the role and use of zero facilitating the exchange and explain the use of Hindu
and uses “debts” to designate development of ideas numerals, the precursors of our
negative quantities. within the Islamic/Arabic world. modern “Arabic” numerals.

C. 628 CE LATE 8TH CENTURY C. 825–830

8TH CENTURY C. 820 C. 930

The spread of Islam into some Al-Khwarizmi writes his Death of Abu Kamil,
parts of India leads to Indian book on algebra, writer of The Book of
mathematicians sharing introducing many methods Algebra, a key influence
their knowledge with for solving equations that for Fibonacci three
Arab scholars. are still important today. centuries later.

A
s the Roman Empire zero is attributed to the brilliant religion had a high regard for
collapsed and Europe mathematician Brahmagupta, who philosophy and scientific enquiry,
entered the Middle described the rules of its use in and the “House of Wisdom”, a
Ages, the centre of scientific and calculation. In fact, the character centre of learning and research
mathematical scholarship shifted may already have been in use for established in Baghdad, attracted
from the eastern Mediterranean to some time. It would have fitted scholars from all over the
China and India. From about the well with India’s numeral system, expanding Islamic Empire.
5th century ce, India began a which is the prototype of our This thirst for knowledge
“Golden Age” of mathematics, modern Hindu–Arabic numerals. prompted the study of ancient
building on its own long tradition Yet it is thanks to Islam that these texts, especially those of the
of scholarship, but also on ideas and other ideas from India’s Golden great Greek philosophers and
brought in by the Greeks. Indian Age (which continued until the mathematicians. Islamic scholars
mathematicians made significant 12th century) went on to influence not only preserved and translated
advances in the fields of geometry the history of mathematics. the ancient Greek texts, but
and trigonometry, which had provided commentaries on them
practical applications in astronomy, Persian powerhouse and developed their own original
navigation, and engineering, but After the death of the Prophet concepts. Open to new ideas,
the most far-reaching innovation Mohammed in 632, Islam rapidly they also adopted many of the
was the development of a character became a major political as well as Indian innovations, in particular
to represent the number zero. religious power in the Middle East their numeral system. The Islamic
The use of a specific symbol – and beyond, spreading from Arabia world, like India, entered a “Golden
a simple circle, rather than a blank across Persia and into Asia as far as Age” of learning that lasted until
space or placeholder – to denote the Indian subcontinent. The new the 14th century, and produced
 THE MIDDLE AGES 87

The binomial theorem,

which makes it possible to The anonymously published
solve equations without Historian Ibn Khallikan Treviso Arithmetic
relying on geometric Robert of Chester makes the first written becomes the first printed
diagrams, is laid translates al-Khwarizmi’s mention of the “wheat on mathematics textbook
out by al-Karaji. work into Latin. a chessboard” problem. in Europe.

C. 1020 1145 1256 1478

C. 1070 1202 14TH CENTURY

Omar Khayyam invents Fibonacci’s Liber Abaci (The Book The Oxford Calculators
a method for of Calculation) introduces many at Merton College give
classifying and ideas from the Arabic world, including the University of Oxford a
solving cubic the Hindu-Arabic numeral system prominent position in
equations. and his famous sequence. Western mathematics.

a succession of influential contact with the Islamic world adopted the Hindu-Arabic numeral
mathematicians – such as increased, and some recognized system, and the use of symbols in
al-Khwarizmi, a key figure in the the wealth of scientific knowledge algebra, and contributed many
development of algebra (the word Islamic scholars had amassed. original ideas, including the
“algebra” derives from the Arabic Christian scholars now gained Fibonacci arithmetical sequence.
term for rejoining), and other access to Greek and Indian With the growth in trade in the
scholars whose contributions to philosophical and mathematical later Middle Ages, mathematics –
the binomial theorem and the texts, and to the work of the Islamic especially the fields of arithmetic
treatment of quadratic and cubic scholars. Al-Khwarizmi’s treatise and algebra – became increasingly
equations were groundbreaking. on algebra was translated into important. Advances in astronomy
Latin in the 12th century by Robert also demanded sophisticated
From East to West of Chester, and soon after, complete calculations. Mathematical
In Europe, mathematical study translations of Euclid’s Elements education was now taken more
was under the control of the and other important texts began seriously. With the invention of the
Church, and was confined to a to appear in Europe. movable-type printing press in
few early translations of some the 15th century, books of all sorts,
of Euclid’s work. Progress was Mathematical renaissance including the Treviso Arithmetic,
hindered by the continued use of City-states in Italy were quick became widely available, spreading
the cumbersome Roman system to trade with the Islamic Empire, the new-found knowledge across
of numerals, necessitating the and it was an Italian, Leonardo Europe. These books inspired a
use of the abacus for calculation. of Pisa, nicknamed Fibonacci, “scientific revolution” that would
However, from the 12th century who spearheaded the revival of accompany the cultural rebirth
onwards, during the Crusades, mathematics in the West. He known as the Renaissance. ■
88
IN CONTEXT

A FORTUNE
KEY FIGURE
Brahmagupta (c. 598–668 ce)
FIELD

SUBTRACTED
Number theory
BEFORE
c. 700 bce On a clay tablet, a
Babylonian scribe indicates

FROM ZERO
a placeholder zero with three
hooks; it is later written as
two slanted wedge marks.
36 bce A shell-shaped zero

IS A DEBT
is recorded on a Mayan stela
(stone slab) in Central America.
c. 300 ce Parts of the Indian
Bakshali text reveal many
circular placeholder zeros.

ZERO AFTER
1202 In his book Liber Abaci,
Leonardo of Pisa (Fibonacci)
introduces zero to Europeans.
17th century Zero is finally
established as a number and
is in widespread use.

A
number that represents
the absence of something
is a difficult concept,
which may be why zero took so
long to become widely accepted.
Several ancient civilizations,
including the Babylonians and the
Sumerians, could claim to have
invented zero, but its use as a
number was pioneered in the
7th century ce, by Brahmagupta,
an Indian mathematician.

The development of zero

Any system for recording numbers
eventually reaches a point at which
it becomes positional; that is to say,
digits are ordered according to their
value to cope with increasingly large
 THE MIDDLE AGES 89
See also: Positional numbers 22–27 ■ Negative numbers 76–79 ■ Binary
numbers 176–77 ■ The law of large numbers 184–85 ■ The complex plane 214–15
Brahmagupta
Born in 598 ce, astronomer and
mathematician Brahmagupta
lived in Bhillamala, northwest
A positive number A negative number India – a centre of learning in
subtracted from zero subtracted from zero those fields. He became head
becomes a negative becomes a positive of the leading astronomical
number. number. observatory at Ujjain, and
incorporated new work on
number theory and algebra
into his studies on astronomy.
Brahmagupta’s use of the
decimal number system and
It is possible to calculate with zero. the algorithms he devised
spread throughout the world
and informed the work of later
mathematicians. His rules
for calculating with positive
and negative numbers, which
he called “fortunes” and
“debts”, are still cited today.
Zero is a number. Brahmagupta died in 668, only
a few years after completing
his second book.

Key works
numbers. All place value (positional) recognizing it as such, which has
systems require a way of denoting been made more difficult by the 628 Brahmasphutasiddhanta
“there is nothing here”. The fact that zero fell in and out of use (The Correctly Established
Babylonians (1894–539 bce), for over time. In about 300 bce, for Doctrine of Brahma)
example, who at first used context example, the Greeks were starting 665 Khandakhadyaka (Morsel
to differentiate between, say, 35 to develop a more sophisticated of Food)
and 305, eventually used a double form of mathematics based on
wedge mark rather like inverted geometry, with quantities being written in the 2nd century ce, the
commas to indicate the empty represented by the lengths of Graeco-Roman scholar Ptolemy
value. In this way, zero entered the lines. There was no need for zero, used a circular symbol positionally
world as a form of punctuation. or indeed negative numbers between digits and at the end of
The problem for historians has (numbers less than 0), as the Greeks a number, but did not consider
been finding evidence for early did not have a positional number it a number in its own right.
civilizations using zero and system (lengths cannot be In Central America, during
nonexistent or negative). the 1st millenium ce, the Mayans
As the Greeks developed the use used a place value system, which
of mathematics in astronomy, they included zero as a numeral, denoted
began to use an “O” to represent by a shell shape. It was one of three
zero, although it is not clear why. In symbols used by the Mayans for
his astronomical manual Almagest, arithmetic; the other two were a
dot representing 1 and a bar for 5.
While the Mayans could calculate
An abax, a table or board covered in
sand, was used by the Greeks to count. up to hundreds of millions, their
Some scholars have suggested that “O” geographical isolation meant that
was used because it was the shape left their mathematics never spread to
when a counter was removed. other cultures. ❯❯
90 ZERO
First-century Indian numerals did not use zero. By the
9th century, Brahmagupta’s zero (highlighted in pink) was
widely used in India, from where it spread via the Arab
India, 1st century ce
world to Europe. There, it met some initial opposition from
Christian religious leaders, who found the concept of zero
satanic because they associated nothingness with the devil.

India, 9th century

Muslim Spain, c. 11th century Arabia, c. 11th century India, c. 11th century

-5 -4 -3 -2 -1 0 1 2 3 4 5
Europe, 15th century Europe, 16th century John Wallis’s number line, England,
17th century

In India, mathematics advanced also described subtracting zero

rapidly in the early centuries of the from both a negative and a positive
1st millennium ce. By the 3rd and number, and noted again that it left
4th centuries, a place value the numbers unchanged.
system had long been in use, and Brahmagupta went on to
by the 7th century – the time of describe the effect of subtracting
Brahmagupta – the use of a circular numbers from zero. He calculated
symbol as a placeholder was already that a positive number subtracted
well established there. from zero becomes a negative
number and a negative number
Zero as a number subtracted from zero becomes a
Brahmagupta established rules for
calculating with zero. He began
by defining it as the result of
subtracting a number from itself –
for example, 3 - 3 = 0. That
established zero as a number in its
own right as opposed to simply a Black holes
figurative notation or placeholder. are where God
He then explored the effect of divided by zero.
calculating with zero. Brahmagupta Steven Wright
The Nadi Yali yantra is part of an showed that if he added zero to a American comedian
18th-century observatory in Ujjain, negative number, the result was
India. A centre of mathematics and
astronomy since Brahmagupta worked equal to that negative number.
there in the 7th century, it lies on the Similarly, adding zero to a positive
intersection of a former zero meridian number produced the same
of longitude and the Tropic of Cancer. positive number. Brahmagupta
 THE MIDDLE AGES 91
positive number. This calculation Hindu–Arabic numbers, which
brought negative numbers into described the place value system
the same number system as including zero. Yet 300 years later
positive numbers. Like zero, when Leonardo of Pisa (better
negative numbers were an abstract known as Fibonacci) introduced
concept rather than positive values Zero is the most magical Hindu–Arabic numerals to Europe,
such as lengths or quantities. number we know. It is the he was still wary of zero and
number we’re striving treated it as an operator like +
Multiplying and dividing towards every day. and - rather than a number.
Brahmagupta went on to examine Bill Gates Even in the 16th century, Italian
zero in relation to multiplication polymath Gerolamo Cardano solved
and described how the product of quadratic and cubic equations
multiplying any number with zero without zero. Europeans finally
is zero, including zero multiplied accepted zero in the 17th century,
by zero. The next step was to when English mathematician
explain division by zero, which John Wallis incorporated zero in
was more problematic. Recording required answer to n/0 is “infinity”, his number line.
the result of dividing a number, but infinity is not a number and
n, by zero as n/0, Brahmagupta cannot be used in calculations. A vital concept
suggested that a number is Dividing zero itself by zero has Mathematics without zero would
unchanged when it is divided proved even trickier. The result mean many of the articles in this
by zero. However, this was later could be zero, if zero divided by book could not have been written:
found to be impossible, as is any number is thought to be zero. there would be no negative
demonstrated by multiplying any It could also be 1, as any number numbers, no coordinate systems,
number by zero (division being divided by itself is 1. no binary systems (and hence no
defined as finding the missing The spread of Islam through computers), no decimals, and
number in a multiplication). parts of India in the 8th century no calculus, because it would
The result cannot be the original led to Indian mathematicians not be possible to describe
number, as any number multiplied sharing their knowledge, including infinitesimally small quantities.
by zero equals zero. the concept of zero, with scholars Advances in engineering would
Mathematicians now describe in the Arab world. In the 9th have been severely restricted. Zero
division by zero as “undefined”. century, the Islamic mathematician is perhaps the most important
Some have suggested that the al-Khwarizmi wrote a treatise on number of all. ■

The Treviso Arithmetic the number system worked. The

unknown author makes 0 the
The figure zero first became 10th number and calls it a “cipher”
known in Italy from the Arte or “nulla” – something that has
dell’ Abbaco (Art of Calculation, no value unless it is written to the
also known as The Treviso right of other numbers to increase
Arithmetic), published their value.
anonymously in 1478 and In the Treviso description,
the first printed mathematics zero is just a placeholder number,
textbook in Europe. It was which itself was still a new notion.
This grid method of multiplication
revolutionary because it was The idea of zero as a number was from the Treviso Arithmetic
written in everyday Venetian not accepted for centuries. It was multiplies the number 56,289 by
for merchants and anyone else also of little interest to the readers 1,234. Zero is used as a placeholder
who wanted to solve calculation of the Arte dell’ Abbaco, most of in the calculation and in the final
problems. It outlined the Hindu– whom wanted to learn how to use solution – 70,072,626. The book
Arabic decimal place value numbers in practical business also illustrated other methods
system and described how calculations in everyday trading. of multiplication.
ALGEBRA IS
AALGEBRA
SCIENTIFIC ART
94 ALGEBRA

IN CONTEXT
KEY FIGURE Algebra deals
Al-Khwarizmi (c. 780–c. 850) with numbers and They are related to
quantities that are things that are known.
FIELD unknown.
Algebra
BEFORE
1650 bce The Egyptian Rhind
papyrus includes solutions to
linear equations.
The determination Unknown quantities
300 bce Euclid’s Elements lays can be determined by
the foundations of geometry. of the unknown
quantities is possible. examining the things
3rd century ce Greek that are known.
mathematician Diophantus
uses symbols to represent
unknown quantities.

T
7th century ce Brahmagupta he origins of algebra – range of general problems. To work
solves the quadratic equation. a mathematical method out lengths and areas, equations
for calculating unknown involving variables (unknown
AFTER quantities – can be traced back quantities) and squared terms
1202 Leonardo of Pisa’s Liber to ancient Babylonians and were devised. Using tables, the
Abaci uses the Hindu-Arabic Egyptians, as equations on Babylonians could also calculate
number system. cuneiform tablets and papyri reveal. volumes, such as the space within
1591 François Viète introduces Algebra evolved from the need to a grain store.
symbolic algebra, in which solve practical problems, often of a
letters are used to abbreviate geometrical nature, requiring the A search for new methods
terms in equations. determination of a length, area, or Over the centuries, as mathematics
volume. Mathematicians gradually developed, problems became longer
developed rules to handle a wider and more complex, and scholars

Al-Khwarizmi Born in c. 780 ce near what is now He wrote a book on geography,

Khiva, Uzbekistan, Muhammad helped construct a world
Ibn Musa al-Khwarizmi moved map, took part in a project to
to Baghdad, where he became a determine the circumference
scholar at the House of Wisdom. of Earth, developed the
Al-Khwarizmi is regarded as astrolabe (an earlier Greek tool
the “father of algebra” for his for navigation), and compiled a
systematic rules for solving linear set of astronomical tables.
and quadratic equations. These Al-Khwarizmi died around 850.
were outlined in his major work
on calculation by “completion and Key works
balancing” – methods he devised
that are still used today. Other c. 820 On the Calculation with
achievements include his text Hindu Numerals
on Hindu numerals, which, in c. 830 The Compendious Book
its Latin translation, introduced on Calculation by Completion
Europe to Hindu-Arabic numerals. and Balancing
 THE MIDDLE AGES 95
See also: Quadratic equations 28–31 ■ The Rhind papyrus 32–33 ■ Diophantine equations 80–81 ■ Cubic equations
102–05 ■ The algebraic resolution of equations 200–01 ■ The fundamental theorem of algebra 204–09

sought new ways to shorten and Islamic world, partly because an equation) could be done by
simplify them. Although early the human form was forbidden in re-joining (al-jabr) – moving
Greek mathematics was largely religious art and architecture, so subtracted terms to the other
geometry-based, Diophantus many Islamic designs were based side of an equation – and then
developed new algebraic methods on geometric patterns. balancing the two sides of the
in the 3rd century ce, and was the Al-Khwarizmi introduced some equation. The word “algebra”
first to use symbols for unknown fundamental algebraic operations, comes from al-jabr.
quantities. However, it would which he described as reduction, Al-Khwarizmi was not working
be more than a thousand years re-joining, and balancing. The in a total vacuum, as he had the
before standard algebraic notation process of reduction (simplifying translated works of earlier Greek ❯❯
was accepted.
After the fall of the Roman
Empire, mathematics in the Key texts in the House of Wisdom
Mediterranean area declined,
but the spread of Islam from the
7th century had a revolutionary
impact on algebra. In 762 ce, Caliph
al-Mansur established a capital in
Baghdad, which swiftly became a
major centre of culture, learning,
and commerce. Its status was
enhanced by the acquisition and
translation of manuscripts from
earlier cultures, including works by
the Greek mathematicians Euclid, Treatise on
Book of Rare
Apollonius, and Diophantus, Demonstration
Things in the
as well as Indian scholars such as of Problems of
Art of Calculation,
Algebra,
Brahmagupta. They were housed Abu Kamil
Omar Khayyam
(c. 850–950 ce)
in a great library, the House of (1070 ce)
Wisdom, which became a centre
for research and the dissemination
of knowledge.
Glorious on Book of
The Compendious
The early algebraists Algebra,
Book on Calculation
Algebra,
Al-Karaji Abu Kamil
Scholars at the House of Wisdom (980–1030 ce) by Completion (850–930 ce)
produced their own research, and and Balancing,
Al-Khwarizmi
in 830, Muhammad Ibn Musa (830 ce)
al-Khwarizmi presented his work
to the library – The Compendious
Book on Calculation by Completion
and Balancing. It revolutionized
ways of calculating algebraic The Correctly
Arithmetica, Elements, Established
problems, introducing principles Diophantus Euclid Doctrine of Brahma,
that are the foundation of modern (3rd century ce) (c. 300 bce) Brahmagupta
algebra. As in earlier periods, the (628 ce)
types of problems discussed were
largely geometrical. The study of
geometry was important in the
96 ALGEBRA
For centuries, algebra had just
It is possible to find x in a been a tool to solve geometric
5 x  8 = 2x + 1 problems, but now became a
linear equation.
discipline in its own right, where
calculating increasingly difficult
equations was the end goal.

Balance the equation by 5x  8 + 8 = 2x +1 + 8 Rational answers

adding the same amount (8) becomes Many of the equations that
to both sides. 5 x = 2x + 9 al-Khwarizmi was dealing with
had solutions that could not be
expressed rationally and completely
using the Hindu-Arabic decimal
Balance the equation system. Although numbers such
5x  2x = 2x  2x + 9 as M2 – the square root of 2 – had
again by subtracting 2x becomes
from both sides. been known since ancient Greek
3x = 9
times and from even earlier
Babylonian clay tablets, in 825 ce,
al-Khwarizmi was the first to make
the distinction between rational
By dividing both sides by 3, numbers – which can be made into
x is revealed. x=3 fractions – and irrational numbers,
which have an indefinite string
of decimals with no recurring
pattern. Al-Khwarizmi described
and Indian mathematicians at his unit. According to al-Khwarizmi, rational numbers as “audible” and
disposal. He introduced the Indian by using his completion and irrational numbers as “inaudible”.
decimal place-value system to the balancing methods, all quadratic Al-Khwarizmi’s work was
Islamic world, which later led to equations – those in which the developed further by Egyptian
the adoption of the Hindu-Arabic highest power of x is x2 – can be mathematician Abu Kamil Shuja
numeral system widely used today. simplified to one of six basic forms. ibn Aslam (c. 850–930 ce), whose
Al-Khwarizmi began by In modern notation, these would Book of Algebra was designed to
studying linear equations, so-called be: ax2 = bx; ax2 = c; ax2 + bx = c;
because they create a straight line ax2 + c = bx; ax2 = bx + c; and
when plotted on a graph. Linear b2 = c. In these six types, the
equations involve only one variable, letters a, b, and c all represent
which is expressed only to the known numbers, and x represents
power of 1, rather than squared or the unknown quantity.
to any higher power. Al-Khwarizmi approached more The principal object of
complex problems too, producing Algebra… is to determine the
Quadratic equations a geometrical method for solving value of quantities which were
Al-Khwarizmi did not employ quadratic equations that used the before unknown… by
symbols; he wrote his equations technique known as “completing considering attentively the
in words, supported by diagrams. the square” (right). He went on to conditions given… expressed
For example, he wrote out the search for a general solution to in known numbers.
equation ( x/3 + 1)( x/4 + 1) = 20 as: cubic equations – in which the Leonhard Euler
“A quantity: I multiplied a third of highest power of x is x3 – but was
it and a dirham by a fourth of it and unable to find one. However, his
a dirham; it becomes twenty,” a pursuit of this goal showed how
dirham being a single coin, used mathematics had progressed since
by al-Khwarizmi to signify a single the time of the ancient Greeks.
 THE MIDDLE AGES 97
Al-Khwarizmi showed how to solve quadratic equations by a
method known as “completing the square”. This example shows
how to find x in the equation x2 + 10x = 39.

x2 + 10x = 39
Algebra is but written x 5 5
geometry and geometry is
but figured algebra. 1. To represent 2, x 2. For 10 ,
x
Sophie Germain draw a square
whose
draw
two identical
French mathematician x sides are , x x+x rectangles
x
making its with sides of
area 2.
x x and 5.
= 39
x 5 5
x
be an academic treatise for other
mathematicians, rather than for
educated people who had a more
5 5
amateur interest. Abu Kamil
embraced irrational numbers as
possible solutions to quadratic x 5
equations, rather than rejecting x 3. Attach the
them as awkward anomalies. In rectangles
his Book of Rare Things in the at the sides
of the square.
Art of Calculation, Abu Kamil x x2 x x x area = 39
attempted to solve indeterminate
equations (those with more than
one solution). He further explored x
this topic in his Book of Birds, in 5
which he posed a miscellany of x 5
bird-related algebra problems, 4. This creates
an additional
including: “How many ways can square, each
one buy 100 birds in the market side of which
with 100 dirhams?”. measures
5 5 52 5 5, making
its area
Geometric solutions area = 25 5  5 = 25.
Up until the era of the Arab x
“algebraists” – from al-Khwarizmi x 5. Add the 25
5
in the 9th century to the death to 39 (the value
of the Moorish mathematician of 2 + 10 )
x x
al-Qalasadi in 1486 – the key x x x x area = to produce
25 + 39 a total area
developments within algebra of 64.
were underpinned by geometrical
representations. For example, x 5
al-Khwarizmi’s method of
“completing the square” in order (x + 5) (x + 5) = 64 6. Each side of the large square
to solve quadratic equations relies (x + 5)2 = 64 is + 5. The area is 64.
x
on consideration of the properties x+5 By finding the square root of 64
= 8 (√64 = 8) then balancing each
of a real square; later scholars x+5-5 = 8-5 side, you discover the value
worked in a similar way. ❯❯ x = 3 of .
x
98 ALGEBRA

Mathematician and poet Omar its academic status. Al-Karaji was Islamic mathematicians gather
Khayyam, for example, was instrumental in this development. in the library of a mosque in an
interested in solving problems He established a set of procedures illustration from a manuscript by
the 12th-century poet and scholar
using the relatively new discipline for performing arithmetic on
Al-Hariri of Basra.
of algebra, but employed both polynomials – expressions that
geometrical and algebraic methods. contain a mixture of algebraic
His Treatise on Demonstration of terms. He created rules for way, and reinforced algebra’s
Problems of Algebra (1070) notably calculating with polynomials, in essential links with arithmetic.
includes a fresh perspective on much the same way that there Mathematical proof is a vital
the difficulties within Euclid’s were rules for adding, subtracting, part of modern algebra and one
postulates, a set of geometric rules or multiplying numbers. This of the tools of proof is called
that are assumed to be true without allowed mathematicians to work mathematical induction. Al-Karaji
requiring a proof. Picking up on on increasingly complex algebraic used a basic form of this principle,
earlier work by al-Karaji, Khayyam expressions in a more uniform whereby he would show an
also develops ideas about binomial algebraic statement to be true
coefficients, which determine how for the simplest case (say n = 1),
many ways there are to select a then use that fact to show that it
number of items from a larger set. must also be true for n = 2 and so
He solved cubic equations, too, on, with the inevitable conclusion
inspired by al-Khwarizmi’s use of that the statement must hold true
Euclid’s geometrical constructions An ounce of algebra is worth for all possible values of n.
for working out quadratic equations. a ton of verbal argument. One of al-Karaji’s successors
John B. S. Haldane was the 12th-century scholar Ibn
Polynomials British mathematical biologist Yahya al-Maghribi al-Samaw’al.
During the 10th and early 11th He noted that the new way of
centuries, a more abstract theory thinking of algebra as a kind of
of algebra was developed, which arithmetic with generalized rules
was not reliant on geometry – an involved the algebraist “operating
important factor in establishing on the unknown using all the
 THE MIDDLE AGES 99
equations and graphs – between variables in an equation: vowels to
mathematical symbols and represent unknown quantities and
visual representations. consonants to represent the known.
Although this convention was
A new algebra eventually replaced by René
As the sun eclipses the stars The discoveries and rules set Descartes – in which letters at
by its brilliancy, so the man of down by medieval Arab scholars the beginning of the alphabet
knowledge will eclipse the still form the basis of algebra represent known numbers and
fame of others in assemblies of today. The works of al-Khwarizmi letters at the end represent the
the people if he proposes and his successors were key unknown – Viète nonetheless
algebraic problems, and still to establishing algebra as a was responsible for simplifying
more if he solves them. discipline in its own right. It was algebraic language far beyond
Brahmagupta not until the 16th century, however, what the Arab scholars had
that mathematicians began to imagined. The innovation allowed
abbreviate equations by using mathematicians to write out
letters to stand for known and increasingly complex and detailed
unknown variables. French abstract equations, without using
mathematician François Viète geometry. Without symbolic
was key to this development. In algebra, it would be difficult to
arithmetical tools, in the same way his works, he pioneered the move imagine how modern mathematics
as the arithmetician operates on away from the Arabic algebra of would have ever developed. ■
the known”. Al-Samaw’al continued procedures towards what is known
al-Karaji’s work on polynomials, but as symbolic algebra.
Islamic algebraists wrote equations
also developed the laws of indices, In his Introduction to the as text with accompanying diagrams, as
which led to much later work on Analytic Arts (1591), Viète in the 14th-century Treatise on the
logarithms and exponentials, and suggested that mathematicians Question of Arithmetic Code by Master
was a significant step forwards should use letters to symbolize the Ala-El-Din Muhammed El Ferjumedhi.
in mathematics.

Plotting equations
Cubic equations had challenged
mathematicians since the time
of Diophantus of Alexandria.
Al-Khwarizmi and Khayyam had
made significant progress in
understanding them – work further
developed by Sharaf al-Din al-Tusi,
a 12th-century scholar, probably
born in Iran, whose mathematics
appears to have been inspired by
the work of earlier Greek scholars,
especially Archimedes. Al-Tusi
was more interested in determining
types of cubic equation than
al-Khwarizmi and Khayyam had
been. He also developed an early
understanding of graphical curves,
articulating the significance of
maximum and minimum values.
His work strengthened the
connection between algebraic
100

FREEING ALGEBRA
FROM THE
CONSTRAINTS
OF GEOMETRY
THE BINOMIAL THEOREM

A
t the heart of many
IN CONTEXT mathematical operations
In ancient Greece, lies an important basic
KEY FIGURE mathematics was almost theorem – the binomial theorem.
Al-Karaji (c. 980–c. 1030) entirely based on geometric It provides a shorthand summary
arguments. of what happens when you
FIELD
Number theory multiply out a binomial, which
is a simple algebraic expression
BEFORE consisting of two known or
c. 250 ce In Arithmetica, unknown terms added together
Diophantus lays down ideas Al-Karaji broke or subtracted. Without the binomial
about algebra later taken up from this tradition and theorem, many mathematical
by al-Karaji. treated the solution of operations would be almost
equations in purely impossible to achieve. The theorem
c. 825 ce The Persian numerical terms. shows that when binomials are
astronomer and mathematician multiplied out, the results follow
al-Khwarizmi develops algebra. a predictable pattern that can be
AFTER written as an algebraic expression
1653 In Traité du triangle or displayed on a triangular grid
arithmétique (Treatise on the (known as Pascal’s triangle after
He created a set of
Arithmetical Triangle), Blaise Blaise Pascal, who explored the
algebraic rules,
Pascal reveals the triangular including the binomial pattern in the 17th century).
pattern of coefficients in the theorem.
Making sense of binomials
bionomial theorem in what is
The binomial pattern was first
later called Pascal’s triangle.
observed by mathematicians in
1665 Isaac Newton develops ancient Greece and India, but the
the general binomial series man credited with its discovery
from the binomial theorem,
Algebraic is the Persian mathematician
forming part of the basis for
solutions no longer al-Karaji, one of many scholars
his work on calculus.
had to rely on who flourished in Baghdad from
geometric diagrams. the 8th to the 14th century.
Al-Karaji explored the multiplication
of algebraic terms. He defined
 THE MIDDLE AGES 101
See also: Positional numbers 22–27 ■ Diophantine equations 80–81 ■ Zero 88–91 ■ Algebra 92–99 ■ Pascal’s triangle
156–61 ■ Probability 162–65 ■ Calculus 168–75 ■ The fundamental theorem of algebra 204–09

1 2 3 4 5
Al-Karaji created a table
to work out the coefficients 1 1 1 1 1
of binomial equations. The first
five lines of it are shown here. 1 2 3 4 5
The top line is for powers, with the
coefficients for each power listed 1 3 6 10 The binomial theorem and a
in the column below. The first and
final numbers are always 1. Each
Bach fugue are, in the long
1 4 10 run, more important than all
other number is the sum of its
adjacent number in the preceding the battles of history.
1 5
column and the number above James Hilton
that adjacent number. British novelist
1
The expansion of (a + b)3
can be found by looking in
the column headed by 3. (a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3

single terms called “monomials”– x, calculated by adding together expressions. The algebra developed
x2, x3, and so on – and showed how pairs of numbers in the preceding by al-Khwarizmi 150 years or
they can be multiplied or divided. column. To determine the powers so previously had used a system
He also looked at “polynomials” in the expansion, you take the of symbols to work out unknown
(expressions with multiple terms), degree of the binomial as n. In quantities and was limited in
such as 6y2 + x3 - x + 17. But it (a + b)2, n = 2. scope. It was tied to the rules
was his discovery of the formula for of geometry, and the solutions
multiplying out binomials that had Algebra breaks free were geometric dimensions,
the most impact. Al-Karaji’s discovery of the binomial such as angles and side lengths.
The binomial theorem concerns theorem helped to open the way for Al-Karaji’s work showed how
powers of binomials. For example, the full development of algebra, by algebra could instead be based
multiplying out the binomial (a + b)2 allowing mathematicians to entirely on numbers, liberating it
by converting it to (a + b) (a + b) manipulate complicated algebraic from geometry. ■
and multiplying each term in the
first bracket by each term in the Al-Karaji Extraction of Hidden Waters
second bracket results in (a + b)2 = is the first known manual
a2 + 2ab + b2. The calculation for Born around 980 ce, Abu Bakr on hydrology.
the power 2 is manageable, but ibn Muhammad ibn al-Husayn Later in life, al-Karaji
for greater powers, the resulting al-Karaji most likely got his moved to “mountain countries”
expression becomes increasingly name from the city of Karaj, (possibly the Elburz mountains
complicated. The binomial theorem near Tehran, but he lived most near Karaj), where he spent
simplifies the problem by unlocking of his life in Baghdad, at the his time working on practical
the pattern in the coefficients – court of the caliph. It was here projects for drilling wells and
numbers, such as 2 in 2ab, by around 1015 that he probably building aqueducts. He died
which the unknown terms are wrote his three key mathematics around 1030 ce.
texts. The work in which
multiplied. As al-Karaji discovered,
al-Karaji developed the binomial Key works
the coefficients can be laid out
theorem is now lost, but later
in a grid, with the columns commentators preserved his Glorious on algebra
showing the coefficients needed ideas. Al-Karaji was also an Wonderful on calculation
for multiplying out each power. engineer, and his book Sufficient on calculation
The coefficients in a column are
102
IN CONTEXT

FOURTEEN
KEY FIGURE
Omar Khayyam (1048–1131)

FORMS WITH ALL

FIELD
Algebra
BEFORE
3rd century bce Archimedes

THEIR BRANCHES
solves cubic equations using
the intersection of two conics.
7th century ce Chinese

AND CASES
scholar Wang Xiaotong solves
a range of cubic equations
numerically.
AFTER
CUBIC EQUATIONS 16th century Mathematicians
in Italy create jealously guarded
methods to solve cubic
equations in the fastest time.
1799–1824 Italian scholar
Paolo Ruffini and Norwegian
mathematician Niels Henrik
Abel show that no algebraic
formulae exist for equations
involving terms to the power
of 5 and higher.

I
n the ancient world, scholars
considered problems in a
geometric way. Simple linear
equations (which describe a line),
such as 4x  8 = 12, where x is
to the power of 1, could be used to
find a length, while a squared
variable (x2) in a quadratic equation
could represent an unknown area –
a two-dimensional space. The next
step up is the cubic equation, where
the x3 term is an unknown volume –
a three-dimensional space.
The Babylonians could solve
quadratic equations in 2000 bce,
but it took another 3,000 years
before Persian poet-scientist Omar
Khayyam found an accurate method
for solving cubic equations, using
 THE MIDDLE AGES 103
See also: Quadratic equations 28–31 ■ Euclid’s Elements 52–57 ■ Conic sections
68–69 ■ Imaginary and complex numbers 128–31 ■ The complex plane 214–15

Cubic equations involve a variable to the

power of 3 (x 3).

The ancient Greeks attempted to solve cubic equations

using only a ruler and compasses.

Omar Khayyam
Born in Nishapur, Persia (now
Omar Khayyam devised more accurate methods for Iran), in 1048, Omar Khayyam
solving cubic equations by: was educated in philosophy
and the sciences. Although he
won renown as an astronomer
and mathematician, when his
patron Sultan Malik Shah died
breaking the in 1092, he was forced into
equation down into a drawing geometric hiding. Finally rehabilitated
simpler equation with diagrams to explore where 20 years later, he lived quietly
squares (powers of 2) and shapes intersect. and died in 1131.
lengths (powers of 1). In mathematics, Khayyam
is best remembered for his
work on cubic equations, but
he also produced an important
commentary on Euclid’s fifth
curves called conic sections – such used a ruler and compasses to postulate, known as the
as circles, ellipses, hyperbolas, attempt constructing a solution to parallel postulate. As an
or parabolas – formed by the this cubic equation but they never astronomer, he helped to
intersection of a plane and a cone. succeeded. Khayyam saw that such construct a highly accurate
tools were not enough to solve all calendar that was used until
Problems with cubes cubic equations, and set out his the 20th century. Ironically,
The ancient Greeks, who used use of conic sections and other Khayyam is now best known
geometry to work out complex methods in his treatise on algebra. for a work of poetry for which
problems, puzzled over cubes. Using modern conventions, he may not have been the sole
A classic conundrum was how to cubic equations can be expressed author – the Rubaiyat, which
was translated into English
produce a cube that was twice simply, such as x3 + bx = c.
by Edward Fitzgerald in 1859.
the volume of another cube. For Without the economy of modern
example, if the sides of a cube are notation, Khayyam expressed
each equal to 1 in length, what his equations in words, describing Key works
length sides do you need for a cube x3 as “cubes”, x2 as “squares”,
twice the volume? In modern terms, x as “lengths”, and numbers as c. 1070 Treatise on
Demonstration of Problems
if a cube with side length 1 has a “amounts”. For example, he
of Algebra
volume of 13, what side length described x3 + 200x = 20x2 + 2,000 1077 Commentaries on the
cubed ( x3) produces twice that as a problem of finding a cube that difficult postulates of
volume; that is, since 13 = 1, what “with two hundred times its side” Euclid’s book
is x if x3 = 2? The ancient Greeks is equal to “twenty squares of ❯❯
104 CUBIC EQUATIONS
its side and two thousand”. For a 144 ⁄ 36
in the example below. The solution would always have been
simpler equation, such as x3 + 36x = circle passed through the origin a positive number. There is an
144, Khayyam’s method was to draw (0,0) and its centre was on the x equally valid negative answer, as
a geometric diagram. He found that axis at (2,0). Using this diagram, shown by the minus numbers in
he could break down the cubic Khayyam drew a perpendicular line the graph below, but although the
equation into two simpler equations: from the point where the circle and concept of negative numbers was
one for a circle, and the other for a parabola intersected down to the recognized in Indian mathematics,
parabola. By working out the value x axis. The point where the line it was not generally accepted until
of x for which both these simpler crossed the x axis (where y = 0) the 17th century.
equations are true simultaneously, gives the value for x in the cubic
he could solve the original cubic equation. In the case of x3 + 36x Khayyam’s contribution
equation. This is shown in the = 144, the answer is x = 3.14 While Archimedes, working in
graph below. At the time, (rounded to two decimal places). the 3rd century bce, may well have
mathematicians did not have these Khayyam did not use coordinates examined the intersection of conic
graphical methods and Khayyam and axes (which were invented sections in a bid to solve cubic
would have constructed the circle about 600 years later). Instead, he equations, what marks Khayyam
and parabola geometrically. would have drawn the shapes as out is his systematic approach. This
Khayyam had also explored the accurately as possible and carefully enabled him to produce a general
properties of conic sections, and measured the lengths on their theory. He extended his mix of
had deduced that a solution to the diagrams. He would then have geometry and algebra to solve cubic
cubic equation could be found by found an approximate numerical equations using circles, hyperbolas
giving the circle in the diagram a solution using trigonometric tables, and ellipses, but never explained
diameter of 4. This measure was which were much used in how he constructed them, simply
arrived at by dividing c by b, or astronomy. For Khayyam, the saying he “used instruments”.
Khayyam was among the first to
realize that a cubic equation could
y have more than one root, and
7 therefore more than one solution.
As can be shown on a modern
6 graph that plots a cubic equation as
a curve snaking above and below
5 the x axis, a cubic equation has up
to three roots. Khayyam suspected
4 two, but would not have considered
parabola x2 = 6y
3

2
G
1
I have shown how to find the
(0,0) H x sides of the square-square,
−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 quatro-cube, cubo-cube…
−1 to any length, which has not
been [done] before now.
−2 Omar Khayyam
circle of diameter 4

A parabola (pink) for the equation x2 = 6 y intersects the circle (blue)

(x2)2 + y2 = 4. A line from G, the point of intersection, to H on the
x axis, gives the value for x (3.14) in the cubic equation x3 + 36x = 144.
 THE MIDDLE AGES 105

Algebras are
geometric facts
which are proved by
propositions.
Omar Khayyam

negative values. He did not like

having to use geometry as well
as algebra to find a solution, and formulae to solve cubic equations. A passion for geometric forms is
hoped that his geometrical efforts Cardano published Ferro’s solution evident in Islamic architecture, seen
would one day be replaced by in his book Ars Magna in 1545. here in the tile patterns, curved arches,
and domes of the Masjid-i Kabud, the
arithmetic. Their solutions were algebraic but
“Blue Mosque”, in Tabriz, Iran.
Khayyam anticipated the differed from those of today, partly
work of 16th-century Italian because zero and negative numbers
mathematicians, who solved cubic were little used at the time. of an “imaginary” unit derived
equations without direct recourse from the square root of a negative
to geometry. Scipione del Ferro Towards modern algebra number, something not possible
produced the first algebraic solution Mathematicians who continued the with “real” numbers. In the late 16th
to cubic equations, discovered in quest for cubic equation solutions century, Frenchman François Viète
his notebook after his death. He included Rafael Bombelli. He was created more modern algebraic
and successors Niccolò Tartaglia, among the first to state that a cubic notation, using substitution and
Lodovico Ferrari, and Gerolamo root could be a complex number, simplifying to reach his solutions.
Cardano all worked on algebraic that is, a number that makes use By 1637, René Descartes had
published a solution to the quartic
The length of the year equation (involving x4), reducing
it to a cubic equation and then to
In 1074, the ruling sultan of visible Sun is directly above two quadratic equations to solve it.
Persia, Jalal al-Din Malik Shah I, the equator. Each month was Today, a cubic equation can be
commissioned Omar Khayyam worked out by the passage of written in the form ax3 + bx2 + cx
to reform the lunar calendar the sun into the corresponding + d = 0, provided a itself is not 0.
used since the 7th century, zodiac region, which required Where the coefficients (a, b, and c,
replacing it with a solar both computations and actual which multiply the variable x) are
calendar. A new observatory observations. Because solar real numbers, rather than complex
was built in the capital Isfahan, transit times could vary by 24 numbers, the equation will have
and Khayyam assembled a team hours, months were between at least one real root and up to
of eight astronomers to assist 29 and 32 days long, but their
three roots in total.
him with the work. length could differ from year to
The year – computed to a
Khayyam’s method is still
year. The new Jalali calendar,
highly accurate 365.24 days – named after the sultan, was taught today. His painstaking work
began at the vernal equinox in adopted on 15 March 1079 and advanced early algebra, while later
March, when the centre of the was only modified in 1925. mathematicians have continued to
refine its expression and scope. ■
THE UBIQUITOUS
MUSIC
OF THE SPHERES
THE FIBONACCI SEQUENCE
108 THE FIBONACCI SEQUENCE

IN CONTEXT Number sequences are lists of numbers linked

KEY FIGURE by a rule.
Leonardo of Pisa, also
known as Fibonacci
(1170–c. 1250)
FIELD In the Fibonacci sequence, starting with 0 and 1,
Number theory the next number is the sum of the previous two.
BEFORE
200 bce The number sequence
later known as the Fibonacci
sequence is cited by the
Indian mathematician The sequence continues 0 + 1 = 1; 1 + 1 = 2;
infinitely. 1 + 2 = 3; 2 + 3 = 5;
Pingala in relation to 3 + 5 = 8; 8 + 5 = 13…
Sanskrit poetic metres.
700 ce The Indian poet and

O
mathematician Virahanka ne sequence of numbers Calculation). The sequence has
writes about the sequence. occurs time and again in important forecasting applications
AFTER the natural world. In this in nature, geometry, and business.
sequence, every number is the sum
17th century In Germany,
of the previous two (0, 1, 1, 2, 3, 5, 8, A problem with rabbits
Johannes Kepler notices
13, 21, 34, and so on). Originally One of the problems Fibonacci
that the ratio of successive referred to by the Indian scholar raised in Liber Abaci concerned
terms in the sequence Pingala in around 200 bce, it was the growth of rabbit populations.
converges. later called the Fibonacci sequence Starting with a single pair of
1891 Édouard Lucas coins after Leonardo Pisano (Leonardo of rabbits, he asked his readers to
the name Fibonacci sequence Pisa), an Italian mathematician work out how many pairs there
in Théorie des Nombres known as Fibonacci. Fibonacci would be in each successive
(Number Theory). explored the sequence in his 1202 month. Fibonacci made several
book Liber Abaci (The Book of assumptions: no rabbit ever died;

Fibonacci Born Leonardo Pisano, probably Leonardo also travelled to

in Pisa, Italy, in 1170, Fibonacci Egypt, Syria, Greece, Sicily, and
did not become known as Provence, exploring different
Fibonacci (“son of Bonacci”) until number systems. His work was
long after his death. Leonardo widely read and came to the
travelled widely with his diplomat attention of the Holy Roman
father and studied at a school of Emperor, Frederick II. Fibonacci
accounting in Bugia, North Africa. died c. 1240–50.
There he came across the Hindu–
Arabic symbols used to represent Key works
the numbers 1 to 9. Impressed
by these numerals’ simplicity 1202 Liber Abaci (The Book
compared with the lengthy Roman of Calculation)
numerals used in Europe, he 1220 Practica Geometriae
discussed them in Liber Abaci (Practical Geometry)
(The Book of Calculation), which 1225 Liber Quadratorum
he wrote in 1202. (The Book of Squares)
 THE MIDDLE AGES 109
See also: Positional numbers 22–27 ■ Pythagoras 36–43 ■ Trigonometry 70–75 ■ Algebra 92–99 ■ The golden ratio 118–23
■ Pascal’s triangle 156–61 ■ Benford’s law 290

January One pair of rabbits,

too young to breed Parent

Offspring
February The rabbit pair is now
mature enough to breed. Each month, some rabbits
mature and others breed.
In the first six months,
March Two rabbit offspring have the number of pairs has
now joined their parents, increased in the sequence
who breed again. 1, 1, 2, 3, 5, and 8. Future
generations over the next
April The second-generation four months can be forecast
rabbits are now old to contain 13, 21, 34, and 55
enough to breed. pairs of rabbits.

May There are now three

mature pairs and two
immature pairs.
June
By month six,
there are eight
pairs of rabbits.

rabbit pairs mated every month, but sequence. As with many three “great grandparents” – its
only after they were two months old, mathematical problems, it is grandmother’s two parents and
the age of maturity; and each pair based on a hypothetical situation: its grandfather’s mother. Further
produced one male and one female Fibonacci’s assumptions about how back, there are five members of
offspring every month. For the first the rabbits behave are unrealistic. the previous generation, eight
two months, he said, there would of the one before that, and so on.
only be the original pair: by the end Generations of bees The pattern is clear: the number
of three months, there would be a An example of the Fibonacci of members in each generation of
total of two pairs; and at the end of sequence cropping up in nature ancestors forms the Fibonacci
four months there would be three concerns bees in a beehive. A male sequence. The sum of the ❯❯
pairs, as only the original pair was bee, or drone, develops from the
old enough to breed. unfertilized egg of a queen bee.
Thereafter, the population Since the egg is unfertilized, the
grows more quickly. In the fifth drone has only one parent, its
month, both the original pair and “mother”. Drones have different
their first offspring produce baby roles in the beehive, one of which is The Fibonacci sequence
rabbits, although the second pair to mate with the queen and fertilize turns out to be the key
of offspring is still too young. This her eggs. Fertilized eggs develop to understanding how
results in a total of five pairs of into female bees, which can either
nature designs.
rabbits. The process continues be queens or workers. This means
in successive months, resulting in that one generation back the drone
Guy Murchie
American writer
a number sequence in which each has only one ancestor, its mother;
number is the sum of the previous two generations back it has two
two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ancestors, or “grandparents” – the
144, and so on – a sequence that mother and father of its mother;
became known as the Fibonacci and three generations back, it has
110 THE FIBONACCI SEQUENCE
six petals, so while numbers from
the sequence are common, other
If a number in a sequence is The ratios of any two
consecutive Fibonacci patterns are also found.
divided by the previous Each Fibonacci number is the
one, it creates a ratio. numbers get closer
and closer to 1.618. sum of the previous two, so the first
two have to be stated before the
third can be calculated. The
Fibonacci sequence can be defined
by a recurrence relation – an
equation that defines a number in
Like the Fibonacci sequence, 1.618 is an approximation of a sequence in terms of its previous
the golden ratio often occurs the “golden ratio”, which
numbers. The first Fibonacci number
in the natural world. is actually (1 + M5)  2.
is written as f 1, the second as f 2, and
so on. The equation is fn = f (n-1) +
f(n-2), where n is greater than 1. If
you are trying to find the fifth
number of parents of a male and a have three, five, or eight petals – Fibonacci number (f 5), for example,
female from the same generation of numbers that belong to the you must add together f4 and f 3.
bees is three. Their parents total five Fibonacci sequence. Ragwort
grandparents, whose own parents flowers have 13 petals, chicory Fibonacci ratios
add up to eight great-grandparents. often has 21, and different types Calculating the ratios of successive
When the pattern is traced back to of daisy have 34 or 55. However, terms in the Fibonacci sequence is
earlier generations, the Fibonacci many other flowers have four or particularly interesting. Dividing
sequence continues, with 13, 21, 34,
55 ancestors, and so on.
The scales of a pine cone, viewed
from above, can be seen to run in two
Plant life Clockwise spirals sets of spirals. Both sets run from the
The Fibonacci sequence can also of scales outside to the centre: one clockwise,
be seen in the arrangement of and the other anticlockwise. The
leaves and seeds in some plants. numbers of spirals in each set
Pine cones and pineapples, for are 13 (clockwise) and 8
(anticlockwise) – two
example, display Fibonacci
Fibonacci numbers.
numbers in the spiral formation of
their exterior scales. Many flowers

[If] a spider climbs so many

feet up a wall each day and
slips back a fixed number each
night, how many days does it
take him to climb the wall?
Fibonacci

Anticlockwise
spirals of scales
 THE MIDDLE AGES 111
A piano keyboard scale from C to
C spans 13 keys, eight white and five
black. The black keys are in groups of
two and three. These numbers all form
part of the Fibonacci sequence.

each number by the previous

number in the sequence produces
the following: 1 ⁄1 = 1, 2 ⁄1 = 2, 3 ⁄2 =
1.5, 5 ⁄3 = 1.666…, 8 ⁄ 5 = 1.6, 13 ⁄8 =
1.625, 21 ⁄13 = 1.61538…, 34 ⁄21 =
1.61904… By continuing this
process indefinitely, it can be
shown that the numbers approach
1.618, approximately. This is
referred to as the golden ratio or the
golden mean. The same number is
also significant in a curve called
the golden spiral, which gets wider
by a factor of 1.618 for every quarter
turn it makes. This spiral crops up
commonly in nature: for example, tradition of 6-line, 20-syllable travers les feuilles (Bells Through
the seeds of pine cones, sunflowers, poetry structured in this way. the Leaves), the ratio of total bars
and coneflowers tend to grow in Around 200 bce, Pingala was aware in the piece to climax bars is
golden spirals. of this pattern in Sanskrit poetry, approximately 1.618.
and the Roman poet Virgil used it Although it is often associated
Arts and analysis in the 1st century bce. with the arts, the Fibonacci
The Fibonacci sequence can also The sequence has also been sequence has also proved a useful
be found in poetry, art, and music. used in music. French composer tool in finance. Today, ratios
A pleasing rhythm in poetry, for Claude Debussy (1862–1918) derived from the sequence are used
example, is produced when employed Fibonacci numbers in as an analytical tool to forecast the
successive lines have 1, 1, 2, 3, 5, several compositions. In the point at which stock market prices
and 8 syllables, and there is a long dramatic climax of his Cloches à will stop rising or falling. ■

Practical solutions several topics in number theory,

including finding Pythagorean
Fibonacci’s work was intended to triples – groups of three integers
have a useful purpose. In Liber that represent the lengths of the
Abaci (1202), for example, he sides of right-angled triangles.
described how to solve many In these triangles, the square
of the problems encountered in of the length of the longest side
commerce, including calculating (the hypotenuse) equals the sum
profit margins and converting of the squares of the lengths of
currencies. In Practica Geometriae the two shorter sides. Fibonacci
(1220), he solved problems found that, starting with 5,
associated with surveying, such every second number in his
as finding the height of a tall object sequence (13, 34, 89, 233, 610
A page from the original using similar triangles (triangles and so on) is the length of the
manuscript of Liber Abaci shows that have identical angles but hypotenuse of a right-angled
the Fibonacci sequence listed different sizes). In his Liber triangle when the lengths of the
on the right. Quadratorum (1225), he tackled two shorter sides are integers.
112

THE POWER
OF DOUBLING
WHEAT ON A CHESSBOARD

T
he first written record of grains and explained the quantity
IN CONTEXT the wheat on a chessboard he desired using the squares on the
problem was made in 1256 8  8 chessboard. One grain of
KEY FIGURE
by Muslim historian Ibn Khallikan, wheat (or rice, in some versions
Sissa ben Dahir
though it is probably a retelling of the story) was to be placed
(3rd or 4th century ce) of an earlier version that arose in on the bottom left square of the
FIELD India in the 5th century. According chessboard. Moving right, the
Number theory to the story, the inventor of chess, number of grains would then be
Sissa ben Dahir, was summoned to doubled, so the second square
BEFORE an audience with his ruler, King had two grains, the third had four,
c. 300 bce Euclid introduces Sharim. The king was so delighted and so on, moving left to right
the concept of a power to with the game of chess that he along each row to the 64th square
describe squares offered to grant Sissa any reward at the top right.
that he wanted. Sissa asked for some Puzzled by what seemed to be a
c. 250 bce Archimedes uses paltry reward, the king ordered that
the law of exponents, which the grains be counted out. The 8th
states that multiplying square had 128 grains, the 24th had
exponents can be achieved more than 8 million, and the 32nd,
by adding the powers. the last square on the chessboard’s
AFTER first half, had over 2 billion. By then,
1798 British economist the king’s granary was running
Thomas Malthus predicts that low, and he realized that the next
square alone, number 33, would
the human population will
need 4 billion grains, or one
grow exponentially while the
large field’s worth. His advisers
food supply will increase more calculated that the final square
slowly, causing a catastrophe. would need 9.2 million trillion
1965 American co-founder of grains, and the total number of
Intel Gordon Moore observes
how the number of transistors Bacteria dividing is an example
on a microchip doubles roughly of exponential growth; when a single
every 18 months. cell divides, it creates two cells that
divide to make four, and so on. This
allows bacteria to spread very quickly.
 THE MIDDLE AGES 113
See also: Zeno’s paradoxes of motion 46–47 ■ Syllogistic logic 50–51
■ Logarithms 138–41 ■ Euler’s number 186–91 ■ Catalan’s conjecture 236–37
The second half of
the chessboard

Sissa’s concept of wheat on a chessboard is an early example

Recent thinkers have used
of how quickly numbers can increase with exponential growth. the chessboard problem as
(Numbers from 1 million onwards are approximate.) The wheat a metaphor for the rate of
on this chessboard would total over 18 million trillion grains. change in technology over
recent years. In 2001,
72 144 288 0.6 1.2 2.3 4.6 9.2 computer scientist Ray
thousand thousand thousand million million million million million Kurzweil wrote an influential
trillion trillion trillion trillion trillion trillion trillion trillion essay describing the
exponential growth in
18 36 technology over previous
281 562 1,123 2,252 4,504 9,007 thousand thousand
trillion trillion trillion trillion trillion trillion trillion trillion years. He predicted that, like
the wheat on the second half
of the chessboard, the rate of
1 2 4 8 17 35 70 140 technological development
trillion trillion trillion trillion trillion trillion trillion trillion would rapidly grow out of
control, following the model of
doubling its previous growth
4 8 16 33 66 131 262 524 with every leap forward.
billion billion billion billion billion billion billion billion
Kurzweil argued that this
rate of growth in technology
would eventually lead to the
16 32 64 128 256 512 1 2 singularity, which is defined
million million million million million million billion billion
in physics as a point at which
a function takes an infinite
65,536 131,072 262,144 524,288 1 million 2 million 4 million 8 million value. When applied to
technology, the singularity
marks the point at which the
cognitive ability of artificial
256 512 1,024 2,048 4,096 8,192 16,384 32,768 intelligence will surpass that
of humans.

1 2 4 8 16 32 64 128 means that 1 will be multiplied

by 2 just once: 1  2  2. The first
square of the chessboard contains
1 grain, so 1 is the first term of
grains on the chessboard would number, the exponent, shows how this series. The number 1 can
be 18,446,744,073,709,551,615 many times the other number, in be written as 20, because it is
(264 – 1). The story has two this case 2, is multiplied by itself. equivalent to 1 multiplied by 2
alternative endings: in one, the The last term in the series, 263, is zero times, leaving 1 unaffected.
king made Sissa his chief adviser; 2 multiplied by itself 63 times. In fact, any number to the power
in the other, Sissa was executed of 0 equals 1 for this reason.
for making the king look foolish. Power of exponents Exponential growth and decay
Sissa’s concept is an example The growth of the values in this relate to many aspects of everyday
of what is known as a geometric series is described as exponential. life. For example, a radioactive
series, in which every successive Exponents can be viewed as isotope decays into another atomic
term is the previous one multiplied instructions for how many times form at an exponential rate, and that
by two: 1  2  4  8  16, and so 1 should be multiplied by a given results in a half-life, where half the
on. From 2 onwards, these numbers number. For example, 23 means material takes the same amount of
are all powers of 2: 1  2  22  23 that 1 will be multiplied by 2 three time to decay, irrespective of the
 24, and so on. The superscript times: 1  2  2  2  8, while 21 starting quantity. ■
THE
RENAISS
1500–1680
ANCE
116 INTRODUCTION

Luca Pacioli Simon Stevin Gilles de Roberval

explores the Robert Recorde makes introduces notation for deduces a method for
golden ratio in his the first use of the non-integer quantities finding the area
Divina Proportione. equals sign (=). to Renaissance Europe. under a cycloid.

1509 1557 1585 1634

1545 1572 1614 1636

Gerolamo Cardano Complex numbers Logarithms are Projective geometry

publishes his Ars are explored invented by John is formulated by
Magna, in which he thoroughly for the first Napier to simplify Girard Desargues.
considers complex time in the Algebra large numbers into
numbers as solutions of Rafael Bombelli. more manageable
to quadratic equations. smaller ones.

T
hroughout the Middle Renaissance art also influenced of symbols to represent functions
Ages, the Catholic Church mathematics. Luca Pacioli, an such as equals, multiplication,
wielded considerable early Renaissance mathematician, and division. Another significant
political power across Europe, and investigated the mathematics development was the formalization
had a virtual monopoly of learning, of the golden ratio that was so of a number system of base-10, and
but in the 15th century its authority important in Classical art, and the Simon Stevin’s introduction of the
was being challenged. A new innovative use of perspective in decimal point in 1585.
cultural movement, known as painting inspired Girard Desargues To meet the era’s practical
the Renaissance (“rebirth”), was to explore the mathematics behind needs, mathematicians devised
inspired by renewed interest in it and develop the field of projective tables of relevant calculations,
the arts and philosophy of the geometry. Practical considerations and John Napier developed a
Graeco-Roman Classical period. also prompted progress: commerce means of calculating with
The Renaissance thirst for required more sophisticated means logarithms in the 17th century.
discovery also accelerated a of accounting, and international The first mechanical aids to
“Scientific Revolution” – classic trade drove advances in navigation, calculation were invented during
texts of mathematics, philosophy, which demanded a deeper this period, such as William
and science had become widely understanding of trigonometry. Oughtred’s slide rule, and Gottfried
available, and inspired a new Leibniz’s mechanical calculating
generation of thinkers. So too Mathematical innovation device, which was a first step
did the Protestant Reformation A major advance in the business of towards true computing devices.
that challenged the hegemony calculation came with the adoption Other mathematicians took a
of the Catholic Church in the of the Hindu-Arabic number more theoretical path, inspired by
16th century. system and an increase in the use the ideas in the newly available
 THE RENAISSANCE 117

Leibniz proposes
The Cartesian system Christiaan Huygens’s a machine that
of coordinates and Blaise Pascal solution to the calculates using binary
axes still in use today publishes his study tautochrone problem principles, laying the
is formalized on the triangle that leads to more foundations for future
by René Descartes. bears his name. accurate clocks. computer coding.

1637 1653 1656 1679

1644 1654 1665–75

The monk Marin The correspondence Calculus is developed

Mersenne describes between Pascal and by Gottfried Leibniz and
the method for Pierre de Fermat lays Isaac Newton, probably
finding the primes the groundwork for independently of
named after him. probability theory. each other.

texts. In the 16th century, the analytic geometry, in which lines scientific giants of the time,
solution of cubic and quartic and shapes are described in terms Gottfried Leibniz and Isaac
equations occupied Italian of algebraic equations. Newton. Following on from the
mathematicians such as Gerolamo Another late-Renaissance work of Gilles de Roberval in
Cardano, while Marin Mersenne mathematician who has become finding the area under a cycloid,
devised a method of finding prime almost a household name is Pierre Leibniz and Newton worked on
numbers, and Rafael Bombelli laid de Fermat, whose claim to fame the problems of calculation of such
down rules for using imaginary rests largely on his enigmatic things as continuous change and
numbers. In the 17th century, the last theorem, which remained acceleration, which had puzzled
pace of mathematical discovery unsolved until 1994. Less well mathematicians ever since
accelerated as never before, known are his contributions to the Zeno of Elea had presented his
and several pioneering modern development of calculus, number famous paradoxes of motion in
mathematicians emerged. Among theory, and analytic geometry. ancient Greece. Their solution
these was philosopher, scientist, He and fellow mathematician to the problem was the theorem
and mathematician René Blaise Pascal corresponded about of calculus, a set of rules for
Descartes, whose methodical gambling and games of chance, calculating using infinitesimals.
approach to problem-solving set laying the foundations for the field For Newton, calculus was a
the scene for the modern scientific of probability. practical tool for his work in physics
era. His major contribution to and especially on the motion of
mathematics was the invention of The birth of calculus planets, but Leibniz recognized
a system of coordinates to specify One of the key mathematical its theoretical importance and
the position of a point in relation to concepts of the 17th century was refined the rules of differentiation
axes, establishing the new field of developed independently by two and integration. ■
THE GEOMETRY
OF ART
AND LIFE
THE GOLDEN RATIO
120 THE GOLDEN RATIO

IN CONTEXT
The golden ratio relates to proportion.
KEY FIGURE
Luca Pacioli (1445–1517)
FIELD
Applied geometry
Two numbers are in a golden ratio if dividing the larger number
BEFORE by the smaller number gives an identical result to dividing the
447–432 bce Designed by the sum of the two numbers by the larger number.
Greek sculptor Phidias, the
Parthenon is later said to
approximate the golden ratio.
c. 300 bce Euclid makes the
first known written reference to Any two consecutive numbers in the Fibonacci sequence,
the golden ratio in his Elements. such as 55 and 89, approximate to a golden ratio.

1202 ce Fibonacci introduces

his famous sequence.
AFTER
1619 Johannes Kepler proves 89  55 = 1.618 144  89 = 1.618
that the numbers in the (rounded down 89 + 55 (rounded up to
Fibonacci sequence approach to three decimal = 144 three decimal
the golden ratio. places) places)
1914 Mark Barr, an American
mathematician, is credited

T
with using the Greek letter he Renaissance was a time mean” – or, as Pacioli called it, the
phi () for the golden ratio. of intellectual creativity, in Divine Proportion – has come to
which disciplines such as symbolize geometrical perfection.
art, philosophy, religion, science, The ratio can be found by dividing
and mathematics were considered a straight line into two parts, so
to be much closer to each other than that the ratio of the longer length
they are today. One area of interest (a) to the smaller length (b) is the
was in the relationship between same as the ratio of the whole line
mathematics, proportion, and (a + b) divided by the longer length
beauty. In 1509, Italian priest and (a). So: (a + b)  a = a  b. The
mathematician Luca Pacioli wrote value of this ratio is a mathematical
Divina Proportione (The Divine constant denoted by the Greek
[The golden proportion] Proportion), which discussed the letter  (“phi”). The name  comes
is a scale of proportions mathematical and geometric from the ancient Greek sculptor
which makes the bad difficult underpinnings of perspective in Phidias (500–432 bce), who is
[to produce] and the architecture and the visual arts. believed to have been one of the
good easy. The book was illustrated by first to recognize the aesthetic
Albert Einstein Pacioli’s friend and colleague possibilities of the golden ratio.
Leonardo da Vinci, a leading artist He allegedly used the ratio in the
and polymath of the Renaissance. design of the Parthenon in Athens.
Since the Renaissance, the Like  (3.1415…),  is an
mathematical analysis of art by irrational number (a number that
means of the “golden ratio”, “golden cannot be expressed as a fraction)
 THE RENAISSANCE 121
See also: Pythagoras 36–43 ■ Irrational numbers 44–45 ■ The Platonic solids 48–49 ■ Euclid’s Elements 52–57
■ Calculating pi 60–65 ■ The Fibonacci sequence 106–11 ■ Logarithms 138–41 ■ The Penrose tile 305

and can therefore be expanded to star, as their symbol. Where one

an infinite number of decimals in side of the pentagram crosses
a non-repeating random pattern. another, it divides each side into
Its approximate value is 1.618. It is two parts, the ratio of which is .
one of the wonders of mathematics The Pythagoreans were convinced
that this seemingly unremarkable that the Universe was based on The good, of course,
number should produce such numbers; they also believed that is always beautiful,
aesthetically pleasing proportions all numbers could be described as and the beautiful
in art, architecture, and nature. the ratio of two integers. According never lacks proportion.
to Pythagorean doctrine, any two Plato
Discovering phi lengths are both integer multiples of
Some believe that proportions some fixed smaller length. In other
related to  can be found in ancient words, their ratio is a rational
Greek architecture – and even number, so it can be expressed as
earlier in ancient Egyptian culture, the ratio of integers. Supposedly,
with the Great Pyramid built at when one of Pythagoras’s followers,
Giza in c. 2560 bce, which has a Hippasus, discovered that this was “extreme and mean ratio”) in
base to height ratio of 1.5717. Yet not true, his fellow Pythagoreans their proportions. Euclid showed
there is no evidence that ancient drowned him in disgust. how to construct the golden ratio
architects were conscious of this using a ruler and compass.
ideal ratio. Approximations to the Written records
golden ratio may have been the The earliest written references Phi and Fibonacci
result of an unconscious tendency to the golden ratio are found in The golden ratio is also closely
rather than any deliberate the work of the Alexandrian related to another well-known
mathematical intention. mathematician Euclid, c. 300 bce. mathematical phenomenon –
The Pythagoreans, a semi- Euclid’s Elements discussed the the set of numbers known as the
mystical group of mathematicians Platonic solids described earlier Fibonacci sequence. It was
and philosophers associated with by Plato (such as the tetrahedron), introduced by Leonardo of Pisa, or
Pythagoras of Samos (570–495 bce) and demonstrated the golden Fibonacci, in his 1202 book Liber
had the pentagram, or five-pointed ratio (which Euclid called the Abaci (The Book of Calculation). ❯❯

Luca Pacioli Luca Pacioli was born in 1445 in students being Leonardo da
Tuscany. After moving to Rome Vinci, who illustrated Pacioli’s
in his youth, he received training Divina Proportione. Pacioli also
from the artist–mathematician devised a method of accounting
Piero della Francesca as well that is still in use today. He died
as the renowned architect Leon in 1517, in Sansepolcro, Tuscany.
Battista Alberti, and gained
knowledge of geometry, artistic
perspective, and architecture. Key works
He became a teacher and travelled
throughout Italy. He also took 1494 Summa de arithmetica,
his vows as a Franciscan friar, geometria, proportioni et
combining monastic pursuits with proportionalita (Summary
teaching. In 1496, Pacioli moved of arithmetic, geometry,
to Milan to work as a payroll proportions, and proportionality)
clerk. While there, he also gave 1509 Divina Proportione
mathematics tuition, one of his (The Divine Proportion)
122 THE GOLDEN RATIO

Leonardo da Vinci supposedly used Another golden ratio approximated the Renaissance in the 15th and
golden rectangles in his composition in nature is the golden spiral, which 16th centuries. Da Vinci’s painting
of The Last Supper (1494–98). Other gets wider by a factor of  for every The Last Supper (1494–98) is said
Renaissance artists – such as Raphael quarter turn it makes. The golden to incorporate the golden ratio. His
and Michelangelo – also used the ratio.
spiral can be drawn by splitting a famous drawing of the “Vitruvian
golden rectangle (a rectangle with Man” – a “perfectly proportioned”
Subsequent numbers in the side lengths in the golden ratio) into man inscribed in a circle and
Fibonacci sequence are found by successively smaller squares and square – for Divina Proportione
adding the previous two together: golden rectangles, and inscribing is also said to contain many
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…. quarter circles inside the squares instances of the golden ratio in
It took until 1619 for German (see opposite). Natural spiral shapes, the proportions of the ideal human
mathematician and astronomer such as the nautilus shell, have a
Johannes Kepler to show that the resemblance to the golden spiral,
golden ratio is revealed if a number but do not strictly fit the proportions.
in the Fibonacci sequence is The golden spiral was first
divided by the one that precedes described by French philosopher,
it. The further along the sequence mathematician, and polymath René
this calculation is attempted, the Descartes in 1638 and was studied The problem with using the
closer the answer is to . For by Swiss mathematician Jacob golden ratio to define human
example, 6,765 ÷ 4,181 = 1.61803. Bernoulli. It was classified as a type beauty is that if you’re looking
Both Fibonacci’s sequence and of “logarithmic spiral” by French hard enough for a pattern,
the golden ratio appear to exist mathematician Pierre Varignon you’ll almost certainly find one.
widely in nature. For example, because the spiral can be Hannah Fry
many species of flower have a generated by a logarithmic curve. British mathematician
Fibonacci number of petals,
and the scales of a pine cone, Art and architecture
viewed from below, are arranged While the golden ratio can be found
in 8 clockwise spirals and 13 in music and poetry, it is more
anticlockwise spirals. often associated with the art of
 THE RENAISSANCE 123
The ratio of beauty
Studies indicate that facial
symmetry plays a major role
in determining a person’s
perceived attractiveness.
However, the proportions
defined by the golden ratio
appear to play an even
greater role. People whose
faces have proportions that
approximate to the golden
ratio (the ratio of the length
of the head to its width, for
instance) are often cited as
being more attractive than
those whose faces do not.
Studies to date, however,
are inconclusive and often
A golden spiral can be inscribed within a golden contradictory; there is little
rectangle. It is created by splitting the rectangle into
scientific basis for believing
squares and a smaller golden rectangle, then repeating
the process in the smaller rectangle. If quarter circles are
that the golden ratio makes
then inscribed in the squares, it creates a golden spiral. a face more attractive.
Stephen Marquardt, an
American plastic surgeon,
body. In reality, the Vitruvian Man, and retrospectively apply the created a “mask” (see below)
which illustrated the theories of golden ratio. Similarly, in 1992, based on applying the golden
ancient Roman architect Vitruvius, American mathematician George ratio to the human face. The
does not quite align with golden Markowsky suggested that more closely a face aligns with
proportions. Despite this, many supposed discoveries of the golden the mask, the more beautiful it
have subsequently attempted to ratio in the human body were a supposedly is. Some, however,
relate the golden ratio to the notion result of imprecise measurements. see the mask – used as a
of attractiveness in people (see template for plastic surgery –
box, right). Modern uses as an unethical, unfounded
use of mathematics.
Although ’s historical use is
Against the golden ratio debated, the golden ratio can still
In the 19th century, German be traced in modern works, such
psychologist Adolf Zeising argued as Salvador Dalí’s Sacrament of the
that the perfect human body Last Supper (1955), in which the
aligned with the golden ratio; it shape of the painting itself is a
could be found by measuring the golden rectangle. Beyond the
person’s total height and dividing arts, the golden ratio has also
this by the height from their feet appeared in modern geometry,
to their navel. In 2015, Stanford particularly in the work of British
mathematics professor Keith Devlin mathematician Roger Penrose,
argued that the golden ratio is a whose Fibonacci tiles incorporate
“150-year scam”. He blamed the golden ratio in their structure.
Zeising’s work for the idea that Standard aspect ratios for television
the golden ratio has historically and computer monitor screens,
had a relationship to aesthetics. such as the 16:9 display, also come The mask created by Stephen
Marquardt has been criticized for
Devlin argues that Zeising’s ideas close to , as do modern bank defining beauty based on white,
have led people to look back at cards, which are almost perfect Western models.
historical art and architecture golden rectangles. ■
124

LIKE A LARGE
DIAMOND
MERSENNE PRIMES

P
rime numbers – numbers interest in the topic, and primes
IN CONTEXT that can only be divided
n
generated by 2 - 1 are now known
by themselves or 1 – have as Mersenne primes ( Mn).
KEY FIGURES
fascinated scholars since the The use of computers has made
Hudalrichus Regius (early
ancient Greeks of Pythagoras’s it possible to find more Mersenne
16th century), Marin
school first studied them, not least primes. Two of Mersenne’s n values
Mersenne (1588–1648) because they can be thought of (67 and 257) were proved incorrect,
FIELD as the building blocks of all natural but in 1947, three new primes were
Number theory numbers (positive integers). Until found: n = 61, 89, and 107 (M61, M89,
1536, mathematicians believed M107), and in 2018, the Great Internet
BEFORE that all prime numbers for n, when Mersenne Prime Search uncovered
c. 300 bce Euclid proves the n
employed in the equation 2 - 1, the 51st known Mersenne prime. ■
fundamental theorem of would lead to another prime as the
arithmetic that every integer solution. However, in his Utriusque
greater than 1 can be Arithmetices Epitome (Epitome of
expressed as a product of Both Arithmetics), published in
primes in only one way. 1536, a scholar known to us only
as Hudalrichus Regius pointed out The beauty of number
c. 200 bce Eratosthenes that 211 - 1 = 2,047. This is not a
devises a method for theory [is] related to the
prime number, as 2,047 = 23  89. contradiction between
calculating prime numbers.
the simplicity of the integers
AFTER Mersenne’s influence and the complicated structure
Regius’s work on primes was
1750 Leonhard Euler confirms
continued by others who proposed
of the primes.
that the Mersenne number n
new hypotheses with 2 - 1. The
Andreas Knauf
231 − 1 is prime. German mathematician
most significant was that of French
1876 French mathematician monk Marin Mersenne in 1644. He
n
Édouard Lucas verifies that stated that 2 -1 was valid when
2127 − 1 is a Mersenne prime. n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127,
and 257. Mersenne’s work rekindled
2018 The largest known
prime to date is found to See also: Euclid’s Elements 52–57 ■ Eratosthenes’ sieve 66–67
be 282,589,933 − 1. ■ The Riemann hypothesis 250–51 ■ The prime number theorem 260–61
 THE RENAISSANCE 125

SAILING ON
A RHUMB
RHUMB LINES

F
rom around 1500, as ships A loxodrome, A great
IN CONTEXT began to cross the world’s or rhumb line circle
oceans, navigators met a
KEY FIGURE
problem – plotting a course across
Pedro Nunes (1502–78)
the world that took account of the
FIELD Earth’s curved surface. The problem
Graph theory was solved by the introduction
of the rhumb line by Portuguese
BEFORE mathematician Pedro Nunes in his
150 ce The Graeco-Roman Treatise on the Sphere (1537).
mathematician Ptolemy
establishes the concepts The rhumb spiral
of latitude and longitude. A rhumb line cuts across every
meridian (line of longitude) at the A meridian,
The angle between
c. 1200 The magnetic compass same angle. Because meridians get
or line of
the rhumb line and
is used by navigators in China, longitude
closer towards the poles, rhumb each line of longitude
Europe, and the Arab world. lines bend around into a spiral. Such
is the same.

1522 Portuguese navigator spirals were called loxodromes by A loxodrome starts at the North or
Dutch mathematician Willebrord South Pole, and spirals around the
Ferdinand Magellan’s ship globe, crossing each meridian at the
completes the first voyage Snell in 1617; they became a key
same angle. A rhumb line is all or part
around the world. concept in the geometry of space. of this spiral.
The rhumb line helps navigators
AFTER because it gives a single compass
1569 Flemish mapmaker bearing for a voyage. In 1569, map. The shortest distance across
Gerardus Mercator’s map Mercator maps – on which lines the globe is not a rhumb, however,
projection allows navigators of longitude are drawn parallel, so but a “great circle” – any circle that
to plot rhumb-line courses as that all rhumb lines are straight – centres on the centre of the Earth.
straight lines on the map. were introduced. This further It only became practical to follow
enabled people to plot a course just a great circle course with the
1617 A spiral rhumb line is by drawing a straight line on the invention of GPS. ■
named a “loxodrome” by Dutch
mathematician Willebrord Snell. See also: Coordinates 144–51 ■ Huygens’s tautochrone curve 167 ■ Graph
theory 194–95 ■ Non-Euclidean geometries 228–29
126

A PAIR OF EQUAL-
LENGTH LINES
THE EQUALS SIGN AND OTHER SYMBOLOGY

I
n the 16th century, when Welsh Widman, but were probably already
IN CONTEXT doctor and mathematician used by German merchants before
Robert Recorde began his work, Widman’s book was published.
KEY FIGURE
there was little consensus on the These symbols slowly replaced the
Robert Recorde (c. 1510–58)
notation used in arithmetic. Hindu– letters “p” for plus and “m” for minus
FIELD Arabic numerals, including zero, as they were taken up by scholars,
Number systems were already established, but there first in Italy, then in England.
was little to represent calculations. In 1557, Recorde went on to
BEFORE In 1543, Recorde’s The Grounde recommend a new symbol of his
250 ce Greek mathematician of Artes introduced the symbols for own. In The Whetstone of Witte,
Diophantus uses symbols to addition (+) and subtraction () to he used a pair of identical parallel
represent variables (unknown mathematics in England. These lines (=) to represent “equals”,
quantities) in Arithmetica. signs had first appeared in print in claiming that “no two things can
Mercantile Arithmetic (1489), by be more equal” than these. Recorde
1478 The Treviso Arithmetic German mathematician Johannes suggested that symbols would save
explains in simple language mathematicians from having to
how to perform addition, write out calculations in words.
subtraction, multiplication, The equals sign was widely adopted,
and division calculations. and the 17th century also saw the
AFTER creation of many of the other
1665 In England, Isaac symbols used today, such as those
Newton develops infinitesimal for multiplication () and division ().
calculus, which introduces
Notating algebra
ideas such as limits, functions,
While the earliest algebraic
and derivatives. These techniques date back more than
processes require new two millennia to the Babylonians,
symbols for abbreviation. most calculations before the
1801 Carl Friedrich Gauss
introduces the symbol for Robert Recorde tested the equals
congruence – equal size sign (=) in his own calculations, as
and shape. seen here in one of his exercise books.
Recorde’s sign was noticeably longer
than the modern form.
 THE RENAISSANCE 127
See also: Positional numbers 22–27 ■ Negative numbers 76–79 ■ Algebra 92–99
■ Decimals 132–37 ■ Logarithms 138–41 ■ Calculus 168–75

The creation of symbols

Symbol Meaning Inventor Date
 Subtraction Johannes Widman 1489
+ Addition Johannes Widman 1489
= Equals Robert Recorde 1557
 Multiplication William Oughtred 1631
< Less than Thomas Harriott 1631 Robert Recorde
> Greater than Thomas Harriott 1631 Born in Tenby, Wales, around
1510, Recorde grew up to
 Division Johann Rahn 1659 study medicine first at Oxford
University, then at Cambridge,
where he qualified as a
16th century were recorded in begin to stick. Descartes also made physician in 1545. He
words – sometimes abbreviated, a contribution with his use of x, y, taught mathematics at both
but not in a uniform way. English and z for the unknowns in universities and wrote the first
mathematician Thomas Harriot equations, and a, b, and c for English book on algebra in
and French mathematician known figures. 1543. In 1549, after a period
practising medicine in London,
François Viète, who each made Algebraic notation may have
Recorde was made controller
important contributions to taken a long time to catch on, but
of the Bristol mint. However,
developments in algebra, used when a symbol made sense and after he refused to issue funds
letters to produce consistent helped mathematicians work to William Herbert, the future
symbolic notation. In their system, through problems, it became the Earl of Pembroke, for his army,
the most noticeable difference from norm. Improved contact between the mint was closed.
today’s notation is the use of a mathematicians in different parts In 1551, Recorde was given
repeated letter to indicate a power. of the world in the 17th century charge of the Dublin mint,
For example, a3 was aaa and also led to such notations being which included silver mines in
x4 was xxxx. adopted much more swiftly. ■ Germany. When he failed to
show a profit, the mines were
A modern system also closed. Recorde later
French mathematician Nicholas tried to sue Pembroke for
Chuquet used superscripts in misconduct, but was instead
1484 to represent exponents (“to the countersued for libel. Sent to
a London prison in 1557 for
power of”), but did not record them
To avoid the tedious repetition failure to pay the fine, Recorde
as such; for example, 6x2 was 6.2. died there in 1558.
It took more than 150 years for of these words, is equal to,
superscripts to become common; I will set, as I do often in
René Descartes used recognizable work use, a pair of parallels. Key works
examples in 1637 when writing 3x Robert Recorde
1543 Arithmetic: or the
+ 5x3, yet continued to write x2 as
xx. Only in the early 19th century, Grounde of Artes
1551 The Pathway to
when the influential German Knowledge
mathematician Carl Gauss favoured 1557 The Whetstone of Witte
using x2, did superscript notation
128
IN CONTEXT

PLUS OF MINUS
KEY FIGURE
Rafael Bombelli (1526–72)

TIMES PLUS
FIELD
Algebra
BEFORE
16th century In Italy,

OF MINUS
Scipione del Ferro, Tartaglia,
Antonio Fior, and Ludovico
Ferrari compete publicly to

MAKES MINUS
solve cubic equations.
1545 Gerolamo Cardano’s
Ars Magna, a book of algebra,
includes the first published
IMAGINARY AND COMPLEX NUMBERS calculation involving
complex numbers.
AFTER
1777 Leonhard Euler introduces
the notation i for 1.
1806 Jean-Robert Argand
publishes a geometrical
interpretation of complex
numbers, leading to the
Argand diagram.

I
n the late 16th century,
Italian mathematician Rafael
Bombelli broke new ground
when he laid down the rules for
using imaginary and complex
numbers in his book Algebra. An
imaginary number, when squared,
produces a negative result, defying
the usual rules that any number
(positive or negative) results in
a positive number when squared.
A complex number is the sum of any
real number (on the number line)
and an imaginary number. Complex
numbers take the form a + bi, where
a and b are real and i = 1.
Over the centuries, scholars
have needed to extend the concept
of the number in order to solve
 THE RENAISSANCE 129
See also: Quadratic equations 28–31 ■ Irrational numbers 44–45 ■ Negative numbers 76–79 ■ Cubic equations 102–05
■ The algebraic resolution of equations 200–01 ■ The fundamental theorem of algebra 204–09 ■ The complex plane 214–15

A real number gives a An imaginary number

positive result when gives a negative result
it is squared. when it is squared.
Some people believe in
imaginary friends. I believe
in imaginary numbers.
R. M. ArceJaeger
American author
A complex number is the sum of a
real number and an imaginary number.

a prestigious university. As a
Complex numbers enable us to solve polynomial result, many mathematicians kept
equations (those that place a sum of powers of x equal to zero, their methods secret rather than
such as 3x3 - 2x2 + x - 5 = 0). sharing them for the common
good. Del Ferro tackled equations
of the form x3 + cx = d. He passed
his technique on to only two people,
different problems. Imaginary and sought to find solutions to cubic Antonio Fior and Annibale della
complex numbers were new tools equations as efficiently as possible, Nave, swearing them to secrecy.
in this endeavour, and Bombelli’s without relying on the geometrical Del Ferro soon had competition
Algebra advanced understanding of methods devised by Persian from Niccolò Fontana (known as
how these and other numbers work. polymath Omar Khayyam in the Tartaglia, or “the stammerer”). An
To solve the simplest equations, 12th century. As most quadratic itinerant teacher of considerable
such as x + 1 = 2, only natural equations could be solved with mathematical ability, but with
numbers (positive integers) are an algebraic formula, the search few financial resources, Tartaglia
needed. To solve x + 2 = 1, however, was on for a similar formula that discovered a general method ❯❯
x must be a negative integer, while worked for cubic equations.
solving x2 + 2 = 1 requires the Scipione del Ferro, a mathematics
square root of a negative number. professor at Bologna University,
This did not exist with the numbers took a major step forward when he
at Bombelli’s disposal, so had to be discovered an algebraic method for
invented – leading to the concept of solving some cubic equations, but
the imaginary unit (-1). Negative the quest for a comprehensive I shall call [the imaginary
numbers were still mistrusted in formula continued. unit] ‘plus of minus’ when
the 1500s; imaginary and complex Italian mathematicians of this added and when subtracted,
numbers were not widely accepted era would publicly challenge one ‘minus of minus’.
for many decades. another to solve cubic equations Rafael Bombelli
and other problems in the least
Fierce rivalry possible time. Achieving fame in
The idea of complex numbers such contests became essential for
first emerged early in Bombelli’s any scholar who wanted to gain a
lifetime as Italian mathematicians post as a mathematics professor at
130 IMAGINARY AND COMPLEX NUMBERS
Bombelli’s rules for the combination of imaginary numbers
Rafael Bombelli set out the rules for operations on positive imaginary unit by a negative imaginary unit, for
complex numbers. He used the term “plus of minus” to example, equals a positive integer; while multiplying a
describe a positive imaginary unit and “minus of minus” negative imaginary unit by a negative imaginary unit
to describe a negative imaginary unit. Multiplying a equals a negative integer.

Plus of minus  Plus of minus = Minus

Plus of minus  Minus of minus = Plus

Minus of minus  Minus of minus = Minus
Minus of minus  Plus of minus = Plus

for solving cubic equations Cardano had to grapple with the significance of this breakthrough
independently of del Ferro. When notion that using the square roots of escaped Cardano; he branded his
del Ferro died in 1526, Fior decided negative numbers might help solve work “subtle” and “useless”.
the time had come for him to cubic equations. He was evidently
unleash del Ferro’s formula upon prepared to experiment with the Explaining the numbers
the world. He challenged Tartaglia method but appears not to have Rafael Bombelli assimilated the
to a cubic duel, but was beaten by been convinced. He called such tussles between the various
Tartaglia’s superior methods. negative solutions “fictitious” and mathematicians solving cubic
Gerolamo Cardano heard of this “false” and described the intellectual equations. He read Cardano’s Ars
and persuaded Tartaglia to share effort involved in finding them as Magna with great admiration. His
his methods with him. As with del “mental torture”. His Ars Magna own work, Algebra, was a more
Ferro, the condition was that the shows his use of the negative accessible version, and was a
method should never be published. square root. He wrote: “Multiply thorough and innovative survey
5 + √15 by 5  √15, making of the subject. It investigated the
Beyond positive numbers 25 (15), which is + 15. Hence arithmetic of negative numbers,
At this time all equations were this product is 40.” This is the and included some economical
solved using positive numbers. first recorded calculation involving notation that represented a major
Working with Tartaglia’s method, complex numbers, but the advance on what had gone before.
The work outlines the basic
Rafael Bombelli his major work, Algebra, which rules for calculating with positive
laid out a primitive but thorough and negative quantities, such as:
Born in Bologna, Italy, in 1526, arithmetic of complex numbers “Plus times plus makes plus; Minus
Rafael Bombelli was the eldest for the first time. times minus makes plus”. It then
of six children; his father was Greatly impressed by a copy sets out new rules for adding,
a wool merchant. Although of Diophantus’s Arithmetica subtracting, and multiplying
Bombelli did not receive a found in the Vatican library, imaginary numbers in terminology
college education, he was taught Bombelli helped to translate it that differs from that used by
by an engineer–architect and into Italian – work that led him mathematicians today. For example,
became an engineer himself, to revise Algebra. Three volumes he stated that “Plus of minus
specializing in hydraulics. He were published in 1572, the year multiplied by plus of minus makes
also developed an interest in he died; the last two incomplete
minus” – meaning a positive
mathematics, studying the work volumes were published in 1929.
imaginary number multiplied by a
of ancient and contemporary
mathematicians. While waiting Key work positive imaginary number equals
for a drainage project to a negative number: √n  √n =
recommence, he embarked on 1572 Algebra n. Bombelli also gave practical
examples of how to apply his rules
 THE RENAISSANCE 131
rational numbers, and irrational
numbers, that were used to solve
equations and perform a range of
other increasingly sophisticated
mathematical tasks.
The shortest route Over the decades, sets of There is an ancient and innate
between two truths in the such numbers acquired their own sense in people that numbers
real domain passes through universal symbols that could be ought not to misbehave.
the complex domain. used in formulae. For instance, the Douglas Hofstadter
Jacques Hadamard bold capital N is used for natural Cognitive scientist
French mathematician numbers from the set {0, 1, 2, 3, 4…},
enclosed in curly brackets to
denote a set. In 1939, American
mathematician Nathan Jacobson
established the bold capital C to
signify the set of complex numbers,
{a + bi}, where a and b are real and function, for example, is a function
for complex numbers to cubic i = 1. of complex numbers that provides
equations, where solutions require Complex numbers enable all information about primes. In other
finding the square root of some polynomial equations to be solved practical areas, physicists use
negative number. Although completely but have also proved complex numbers in the study of
Bombelli’s notation was advanced immensely useful in many other electromagnetism, fluid dynamics,
for his time, the use of algebraic branches of mathematics – even and quantum mechanics, while
symbols was still in its infancy. in number theory (the study of engineers need them for designing
Two centuries later, Swiss integers, especially positive electronic circuits, and for studying
mathematician Leonhard Euler numbers). By treating the integers audio signals. ■
introduced the symbol i to denote as complex numbers (the sum of
the imaginary unit. a real value and an imaginary
value), number theorists can use A series of cups shows blue food
dye being dripped over an ice cube
Applying complex numbers powerful techniques of complex (left). As the ice cube melts, the heavier
Imaginary and complex numbers analysis (a study of functions with blue dye sinks. Complex numbers are
joined the ranks of other sets, such complex numbers) to investigate used to model the velocity (speed
as natural numbers, real numbers, the integers. The Riemann zeta and direction) of such fluids.
THE ART OF
TENTHS
DECIMALS
134 DECIMALS

F
ractions – so named for the
IN CONTEXT Latin word fractio, meaning
“break” – were used from
KEY FIGURE
around 1800 bce in Egypt to express
Simon Stevin (1548–1620)
parts of a whole. At first they were
FIELD limited to unit fractions, which By relieving the brain
Number systems are those with a 1 as the numerator of all unnecessary work,
(top number). The ancient Egyptians a good notation sets it free
BEFORE had symbols for 2 ⁄3 and 3 ⁄4, but other to concentrate on more
830 ce Al-Kindi’s four-volume fractions were expressed as the advanced problems.
On the use of Indian numerals sum of unit fractions, for example Alfred North Whitehead
spreads the place value system as 1 ⁄3 + 1 ⁄13 + 1 ⁄17. This system British mathematician
based on the Hindu numerals worked well for recording amounts
throughout the Arab world. but not for doing calculations. It
was not until after Simon Stevin’s
1202 Leonardo of Pisa’s De Thiende (The Art of Tenths) was
Liber Abaci (The Book of published in 1585 that a decimal
Calculation) brings the Arabic system became commonplace.
number system to Europe. on a system of twelfths, and
AFTER The importance of 10 written out in words: 1 ⁄12 was
1799 The metric system Simon Stevin, a Flemish engineer called uncia, 6 ⁄12 was semis, and
and mathematician in the late 1 ⁄24 was semiuncia, but this
is introduced for French
16th and early 17th century, used cumbersome system made it
currency and measures
many calculations in his work. difficult for people to do any
during the French Revolution.
He simplified these by using calculations. In Babylon, fractions
1971 Britain introduces fractions with a base system of were expressed using their base-60
decimalization, dispensing tenth powers. Stevin correctly number system, but in writing, it
with pounds, shillings, and predicted that a decimal system was difficult to distinguish which
pence, which stemmed from would eventually be universal. numbers represented integers and
the Latin system. Cultures throughout history which were part of the whole.
had used many different bases for For many centuries, Europeans
expressing parts of a whole. In used Roman numerals to record
ancient Rome, fractions were based numbers and to do calculations.

Simon Stevin Born in 1548 in Bruges, now in was responsible for several
Belgium, Simon Stevin worked as innovative military and
a book-keeper, cashier, and clerk engineering ideas that were
before entering the University of adopted across Europe. He
Leiden in 1583. There he met authored many books on a
Prince Maurice, the heir of William variety of subjects, including
of Orange, and they became mathematics. He died in 1620.
friends. Stevin tutored the prince
in mathematics and also advised Key works
him on military strategy, leading
to some significant victories over 1583 Problemata geometrica
the Spanish. In 1600, Prince (Geometric Problems)
Maurice asked Stevin, who was 1585 De Thiende (The Art
also an outstanding engineer, to of Tenths)
found a School of Engineering at 1585 De Beghinselen der
the University in 1600. As quarter- Weeghconst (Principles of
master general from 1604, Stevin the Art of Weighing)
 THE RENAISSANCE 135
See also: Positional numbers 22–27 ■ Irrational numbers 44–45 ■ Negative numbers 76–79 ■ The Fibonacci sequence
106–11 ■ Binary numbers 176–77

Medieval Italian mathematician

Leonardo of Pisa (also known as
To decimalize, fractions are converted to decimal fractions,
Fibonacci) came across the where the denominator (bottom number) is a power of 10.
Indian place-value number system
while he was travelling in the
Arab world. He quickly realized
its usefulness and efficiency for
both recording and calculating
with whole numbers. His Liber The numerator (top number) of the converted fraction is used to
Abaci (1202), which brought many write the fraction as a decimal – for example, 25⁄100 becomes 0.25.
useful Arabic ideas to the west,
also introduced a new notation for
fractions to Europe that would
form the basis of the notation
used today. Fibonacci employed
a horizontal bar to divide the The numerator is placed after a decimal separator, such as
numerator and denominator a decimal point, to show that it is not a whole number.
(bottom number), but followed the
Arabic practice of writing the
fraction to the left of the integer,
rather than to the right.
The decimal system makes it easy to add and subtract
Introducing decimals amounts that are not whole numbers.
Finding that conventional fractions
were both time-consuming and
prone to errors, Stevin began using
a decimal system. The idea of was Stevin who made decimals 6 ⁄100 + 7⁄1,000 – could now be written

“decimal fractions” – which have commonplace in Europe, both for as a single number. Stevin placed
powers of 10 as the denominator – recording and calculating with circles after each number; these
had been used five centuries before parts of a whole. He suggested were shorthand for the denominator
Stevin, in the Middle East, but it a notation system for decimal of the original decimal fraction.
fractions, replicating the The whole 32 would be followed
advantages of the Indian place- by a 0, because 32 is an integer,
value system for whole numbers. whereas the 6 ⁄100, for example,
In Stevin’s new notation, was expressed as 6 and a 2 inside
numbers that would previously a circle. This 2 denoted the power
have been written as the sum of of 10 of the original denominator,
Decimals [are] a kind fractions – for example, 32 + 5 ⁄10 + as 100 is 102. In the same vein, ❯❯
of arithmetic invented
by the tenth progression,
Stevin’s notation used circles to indicate the power of ten of
consisting in characters the denominator of the converted fraction. This represents how
of cyphers. Stevin would have written the number now expressed as 32.567.
Simon Stevin

320516273
136 DECIMALS
The decimal system makes it easier to divide and multiply fractions, especially by
10. Shown here with the example of 32.567 (or 32 + 5/10 + 6 ⁄100 + 7⁄1,000), numbers shift
one column to the left or right, crossing over the decimal separator.

Hundreds Tens Units Tenths Hundredths Thousandths Ten-thousandths

100 10 1 1 1 1 1
10 100 1,000 10,000

×1 3 2 5 6 7

× 10 3 2 5 6 7

÷ 10 3 2 5 6 7

the 7⁄1,000 became a 7 followed in the UK, the commas in the wrong. In an attempt to solve
by a 3 inside a circle. The entire number 2,500,000 are delimiters this problem, the 22nd General
sum could be written out following and are used to make it easier both Conference on weights and
this pattern (see p.135, bottom to read the number and to recognize measures – a meeting of delegates
right). The symbol that is placed its size. The UK uses a point for the from 60 nations of the International
between the whole-number part decimal separator and a comma as Bureau of Weights and Measures –
and the fractional part of a number a delimiter. Elsewhere in the world, decided in 2003 that, although
is called the decimal separator. if a comma is used for the decimal either a point or comma on the
Stevin’s zero inside a circle later separator, a point is then used as line could be used as the decimal
evolved into a dot, now called the delimiter. In Vietnam, for separator, the delimiter was to be
the decimal point. The dot was example, a price of two hundred
positioned on the midline (at a thousand Vietnamese dong is
In Spain, the decimal separator is
middle height) in Stevin’s notation often written as 200.000. a comma, as seen in the prices at this
but has now moved to be on the Usually, the context is sufficient market stall in Catalonia. In handwritten
baseline to avoid confusion with for people to interpret the notation Spanish, an upper comma (similar to
the dot notation sometimes used correctly, but this can go badly an apostrophe) is also common.
for multiplication. Stevin’s circled
numbers for tenth powers were
also done away with, meaning that
32 + 5 ⁄10 + 6 ⁄100 + 7⁄1,000 could now
be written as 32.567.

Different systems
The decimal point has never
become universally accepted.
Many countries use a comma as
the decimal separator instead of a
point. There would be no problem
with the two common notations
if not for the use of delimiters –
symbols that separate groups of
three digits in the whole-number
section of a very large or sometimes
very small number. For example,
 THE RENAISSANCE 137
Terminating and
recurring decimals
Fractions are converted to
decimals by dividing the
numerator by the denominator.
If the denominator is only
divisible by 2 or 5 and no other
prime numbers – as is the case
for 10 – then the decimal will
terminate. For example, 3 ⁄40
can be expressed as 0.075, and
this value is exact because 40
is only divisible by the primes
2 and 5.
Other fractions become
recurring decimals – meaning
that they do not end. For
a space rather than either of the This marble plaque on the rue de example, 2 ⁄11 is decimalized as
Vaugirard, Paris, is one of 16 original 0.18181818…, denoted as 0.e 1e 8
previous symbols. This notation
metre markers installed in 1791, after to show that both the 1 and
is yet to become universal.
the French Académie des Sciences 8 recur. The length of the
defined the metre for the first time. recurring cycle (two numbers
Benefits of decimals in the case of 0.e 1e 8) can be
The same processes of addition, predicted as it will be a factor
subtraction, multiplication, and France also tried to introduce a of the denominator minus 1
division of whole numbers can decimal system for time; there (so if the denominator of the
be used with decimal numbers, would be 10 hours in a day, 100 fraction is 11, the number of
resulting in a far simpler way of minutes in each hour, and 100 digits in the cycle is a factor of
performing basic arithmetic than seconds in each minute. The 10). These differ from irrational
the previous method, which relied attempt was so unpopular that it numbers, which do not
on learning a different set of rules for was dropped after just one year. terminate and have no pattern
calculations with fractions. When The Chinese had introduced of recurrence. Irrational
multiplying fractions, for example, various forms of decimal time numbers cannot be expressed
the numerators would be multiplied over some 3,000 years, but finally as a fraction of two integers.
separately from the denominators, abandoned it in 1645 ce.
and the resulting fraction would In the US, the use of a decimal
then be reduced. With decimal system for measurement and
fractions, multiplying and dividing coinage was championed by
by powers of 10 is straightforward – Thomas Jefferson. His 1784 paper
as in the example of 32.567 (see top persuaded Congress to introduce
left), the decimal separator can be a decimal system for money using
simply moved left or right. dollars, dimes, and cents. In fact, Perhaps the most important
Stevin believed that the the name “dime” originates from event in the history of
universal introduction of decimal Disme, the French title of The Art science… [is] the invention
coinage, weights, and measures of Tenths. Yet Jefferson’s view did of the decimal system…
would only be a matter of time. The not hold sway for measurement, Henri Lebesgue
introduction of decimal measures and inches, feet, and yards are still French mathematician
for length and weight (using metres used today. While many European
and kilograms) arrived in Europe currencies were decimalized in the
some 200 years later, during the 19th century, it was not until 1971
French Revolution. When it that decimal currency was
introduced the metric system, introduced in the UK. ■
138
IN CONTEXT

TRANSFORMING
KEY FIGURE
John Napier (1550–1617)

MULTIPLICATION
FIELD
Number systems
BEFORE
14th century The Indian

INTO ADDITION
mathematician Madhava of
Kerala constructs an accurate
table of trigonometric sines to
aid calculation of angles in
LOGARITHMS right-angled triangles.
1484 In France, Nicolas
Chuquet writes an article
about calculation using
geometric series.
AFTER
1622 English mathematician
and clergyman William
Oughtred invents the slide
rule using logarithmic scales.
1668 In Logarithmo-technia,
German mathematician
Nicholas Mercator first uses
the term “natural logarithms”.

F
or thousands of years, most
calculations were carried
out by hand, using devices
such as counting boards or the
abacus. Multiplication was
especially long-winded and much
more difficult than addition. In the
scientific revolution of the 16th and
17th centuries, the lack of a reliable
calculating tool hampered progress
in areas such as navigation and
astronomy, where the potential for
error was greater because of the
lengthy calculations involved.

Solving by series
In the 15th century, French
mathematician Nicolas Chuquet
investigated how the relationships
 THE RENAISSANCE 139
See also: Wheat on a chessboard 112–13 ■ The problem of maxima 142–43
■ Euler’s number 186–91 ■ The prime number theorem 260–61

In the 16th century, multiplying large numbers was

a long and laborious process.

In these tables, each

John Napier simplified the
number had its equivalent
process by developing tables
“artificial number”, or
of numbers.
logarithm.
John Napier
Born into a wealthy family
in 1550 at Merchiston Castle,
near Edinburgh, John Napier
Two numbers could be
Logarithms enable would later become 8th Laird
multiplied by adding their
mathematicians to complete of Merchiston. Aged just 13, he
logarithms together and
complex multiplications entered St Andrews University
converting the result back
through addition. and became passionately
to an ordinary number.
interested in theology. Before
graduating, however, he left to
study in Europe, although few
details of this time are known.
between arithmetic and geometric an arithmetic sequence (such as
Napier returned to Scotland
sequences could aid calculation. 1, 2, 3, 4…), it can be seen that the
in 1571 and devoted much
In an arithmetic sequence, each top numbers are the exponents to time to his estates, where
number differs from the one which 2 is raised to arrive at the he devised new methods of
preceding it by a constant quantity, series below. It was a much more agriculture to improve his
such as 1, 2, 3, 4, 5, 6… (going up sophisticated version of this land and livestock. A fervent
by 1), or 3, 6, 9, 12… (going up by scheme that lay at the heart of the Protestant, he also wrote a
3). In a geometric sequence, each tables of logarithms developed by prominent book attacking
number after the first term is Scottish landowner John Napier. Catholicism. His keen interest
determined by multiplying the in astronomy, and a desire to
previous number by a fixed amount, Generating logarithms find simpler ways to perform
called the “common ratio”. For Napier was fascinated by the calculations that it
example, the sequence 1, 2, 4, 8, numbers and spent much of his required, led to his invention
16 has a common ratio of 2. Setting time finding ways of making of logarithms. He also created
down a geometric sequence calculations easier. In 1614, he Napier’s Bones, a calculation
device using numbered rods.
(such as 1, 2, 4, 8…) and above it published the first description ❯❯
Napier died at Merchiston
Castle in 1617.
 
0 1 2 3 4 5 6 7 8 9 10 Key works
1 2 4 8 16 32 64 128 256 512 1024
  1614 Mirifici Logarithmorum
The lower row of this table is a geometric sequence (progressing powers Canonis Descriptio (A
of 2), while the top row is an arithmetic sequence that reveals the exponents Description of the Marvellous
(powers) by which 2 is raised to arrive at the numbers in the lower row. Rule of Logarithms)
(Anything to the power of 0 is 1.) To multiply the numbers 16 and 32 in the 1617 Rabdologiae
lower row, their exponents (4 + 5) can be added together to produce 2 9 (= 512).
140 LOGARITHMS
and table of logarithms; a logarithm infinite line, will never arrive at the
of a given number is the exponent end of its journey. At any instant
or power to which another fixed there is a unique correspondence
number (the base) is raised to between the positions of the
produce that given number. The use two particles. The distance
of such tables facilitated complex I found at the first particle has travelled
calculations and advanced the length some excellent is the logarithm of the distance
development of trigonometry. brief rules. the second particle has yet to
As Napier recognized, the basic John Napier go. The first particle’s progress
principle of calculating was simple can be viewed as an arithmetic
enough: he could replace the progression, while that of the
tedious task of multiplication by second particle is geometric.
the simpler operation of addition.
Each number would have its Improving the method
equivalent “artificial number” as It took Napier 20 years to complete
he initially termed it. (Napier later starting position at the same time his calculations and to publish his
settled on the name “logarithm”, and at the same velocity. The first logarithm tables as Mirifici
derived by combining the Greek particle on the infinite line travelled Logarithmorum Canonis Descriptio
words logos, meaning proportion, with uniform motion, so it covered (A Description of the Marvellous
and arithmos, meaning number.) equal distances in equal times. Rule of Logarithms). Henry Briggs,
Adding the two logarithms, and The velocity of the second particle professor of mathematics at the
then converting the answer back to was proportional to the distance University of Oxford, recognized
an ordinary number, produces the remaining to the end of the line. the significance of Napier’s tables
result of multiplying the original Halfway between the starting point but thought them unwieldy.
numbers. For division, one logarithm and the end of the line, the second Briggs visited Napier in 1616
is subtracted from another and the particle is travelling at half the and again in 1617. Following their
result is converted back. velocity it started with; at the discussions, the two agreed that
To generate his logarithms, three-quarter point, it is travelling the logarithm of 1 should be
Napier imagined two particles with a quarter of its initial velocity; redefined as 0 and the logarithm
travelling along two parallel lines. and so on. This means that the of 10 as 1. This approach made
The first line was of infinite length, second particle is never going to logarithms much easier to use.
while the second was of fixed reach the end of the line, and Briggs also helped with the
length. Each particle left the same equally, the first particle, on its calculation of logarithms of

Logarithmic scales
When measuring physical In acoustics, sound intensity
variables, such as sound, flow, is measured in decibels. The
or pressure, where values may decibel scale takes the hearing
change exponentially, rather threshold, defined as 0 dB, as
than by regular increments, a its reference level. A sound 10
logarithmic scale is often used. times louder is assigned a
Such scales use the logarithm of decibel value of 10; a sound 100
a value instead of the actual value times louder has a decibel value
of whatever is being measured. of 20; a sound 1,000 times louder
Each step on a logarithmic scale a value of 30, and so on. This
is a multiple of the preceding logarithmic scale fits well with
The pH logarithmic scale measures step. For example, on a log10 scale, the way we hear things, as a
alkalinity and acidity. A pH of 2 is 10 every unit up the scale represents sound must become 10 times
times more acidic than a pH of 3 and a 10-fold increase in whatever is more intense to sound twice
100 times more acidic than pH 4. being measured. as loud to the human ear.
 THE RENAISSANCE 141
for a planet to complete one
orbit of the Sun is related to its
average orbital distance. When
he published this finding in
1620 in his book Ephemerides
novae motuum coelestium, Kepler By shortening the labours,
dedicated it to Napier. [Napier] doubled the life of
the astronomer.
The exponential function Pierre-Simon Laplace
Later in the 17th century, logarithms
revealed something of further
significance. While studying number
series, Italian mathematician Pietro
Mengoli showed that the alternating
series 1  ½  1 ⁄3  1 ⁄4  1 ⁄5 …
has a value of around 0.693147, such as finance and statistics,
which he demonstrated to be the and most areas of science. The
natural logarithm of 2. A natural exponential function is given in
x
logarithm (ln) – so-called because the form f ( x) = b , where b is
it occurs naturally, revealing the greater than 0 but does not equal
time required to reach a certain 1, and x can be any real number.
Napier’s book describing logarithms level of growth – has a special In mathematical terms, logarithms
was published in 1614, as its title page base, later known as e, with an are the inverse of exponentials
shows. The principles behind his approximate value of 2.71828. This (powers of a number) and can be
logarithm tables were published in number is hugely significant in to any base.
1619, two years after his death.
mathematics due to its links with
natural growth and decay. A basis for Euler’s work
ordinary numbers based on the It was through work such as The push for accurate log tables
logarithm of 10 being 1 and spent that of Mengoli that the important spurred mathematicians such as
several years recalculating the concept of the exponential function Nicholas Mercator to pursue further
tables. The results were published came to light. This function is research in this area. In Logarithmo-
in 1624 with the logarithms used to represent exponential technica, published in 1668, he set
calculated to 14 decimal places. growth – where the rate of growth out a series formula for the natural
The base-10 logarithms calculated of a quantity is proportional to its logarithm ln(1  x) = x  x2/2 + x3/3
by Briggs are known as log10 or size at any particular moment,  x4 /4 … This was an extension
common logarithms. The earlier so the bigger it is, the faster it of Mengoli’s formulation, in which
table to the power of 2 (see p.139) grows – which is relevant to fields the value of x was 1. In 1744, more
can be thought of as a simple than 130 years after Napier
base-2, or log2 table. produced his first logarithm table,
Swiss mathematician Leonhard
The impact of logarithms Euler published a full treatment
Logarithms had an immediate of e and its relationship to the
x
impact on science, and on natural logarithm. ■
astronomy in particular. German
astronomer Johannes Kepler had
published his first two laws of The slide rule, used here in 1941 by
planetary motion in 1605, but only a member of the Women’s Auxiliary
Air Force, is marked with logarithmic
after the invention of log tables was scales that facilitate multiplication,
he able to make the breakthrough division, and other functions. Invented
to discover his third law. This in 1622, it was a vital mathematical tool
describes how the time it takes before the advent of pocket calculators.
142

NATURE USES AS
LITTLE AS POSSIBLE
OF ANYTHING
THE PROBLEM OF MAXIMA

A
stronomer Johannes Kepler
IN CONTEXT is best known for his Kepler
discovery of the elliptical felt cheated by
KEY FIGURE wine merchants and
shape of the planets’ orbits and his
Johannes Kepler (1571–1630) wanted an accurate way to
three laws of planetary motion, but
FIELD he also made a major contribution measure a barrel’s
to mathematics. In 1615, he devised contents.
Geometry
a way of working out the maximum
BEFORE volumes of solids with curved
c. 240 bce In Method of shapes, such as barrels.
Mechanical Theorems, Kepler’s interest in this field Inspired
Archimedes uses indivisibles began in 1613, when he married his by Archimedes, he
to estimate the areas and second wife. He was intrigued used the method of
volumes of curvilinear shapes. when the wine merchant at the infinitesimals to divide the
wedding feast measured the wine barrel into thin sections and
AFTER in the barrel by sticking a rod find the exact volume
1638 Pierre de Fermat diagonally through a hole in the top of wine.
circulates his Method for and checking how far up the stick
determining Maxima and the wine went. Kepler wondered
Minima and Tangents for whether this worked equally well for
Curved Lines. all shapes of barrel and, concerned
that he may have been cheated, The method Kepler used
1671 In Treatise on the was a key step in the
decided to analyse the issue of
Method of Series and Fluxions, development of calculus.
volumes. In 1615, he published his
Isaac Newton produces new
results in Nova stereometria
analytical methods for solving
doliorum vinariorum (New solid
problems such as the maxima geometry of wine barrels).
and minima of functions. Kepler looked at ways of fitted into any shape and added
1684 Gottfried Leibniz calculating the areas and volumes up. The area of a circle could be
publishes New Method for of curved shapes. Since ancient determined, for example, by
Maximums and Minimums, times, mathematicians had using slender pie-slice triangles.
his first work on calculus. discussed using “indivisibles” – To find the volume of a barrel
elements so tiny they cannot be or any other 3-D shape, Kepler
divided. In theory these can be imagined it as a stack of thin
 THE RENAISSANCE 143
See also: Euclid’s Elements 52–57 ■ Calculating pi 60–65 ■ Trigonometry 70–75
■ Coordinates 144–51 ■ Calculus 168–75 ■ Newton’s laws of motion 182–83

Barrel 1
Bung hole in the
The merchant’s rod centre of the barrel

50
is submerged to an

cm
Distance between
equal extent when the bung hole and
pushed at a diagonal the opposite edge
into these two barrels,
so he charges the same Merchant’s rod
price for both. However,
the elongated shape
of the second barrel
means it has a smaller Barrel 2 Johannes Kepler
volume, containing less
Bung hole in the
wine but for the same centre of the barrel Born near Stuttgart, Germany,
price as the first. in 1571, Johannes Kepler
50 Distance between witnessed the “Great Comet”
cm
the bung hole and of 1577 and a lunar eclipse,
the opposite edge and remained interested in
astronomy throughout his life.
Merchant’s rod Kepler taught at the
Protestant seminary in Graz,
layers. The total volume is the sum changing the barrel height would Austria. In 1600, non-Catholics
were expelled from Graz and
of the volumes of the layers. In a change its volume. It turned out
Kepler moved to Prague,
barrel, for example, each layer is that the maximum volume is held
where his friend Tycho Brahe
a shallow cylinder. in short, squat barrels with a height lived. Following the death
just under 1.5 times the diameter – of his first wife and son, he
Infinitesimals like the barrels at his wedding. In moved to Linz in Austria,
The problem with cylinders is contrast, the tall barrels from where his main job as imperial
that if they have thickness, their Kepler’s homeland on the River mathematician was to make
straight sides will not fit into the Rhine held much less wine. astronomical tables.
curve of a barrel, while cylinders Kepler also noticed that the Kepler was convinced that
without thickness have no volume. nearer to the maximum the shape God had made the Universe
Kepler’s solution was to accept the gets, the less the rate at which the according to a mathematical
notion of “infinitesimals” – the volume increases: an observation plan. He is best known for his
thinnest slices that can exist that contributed to the birth of work in astronomy, especially
without vanishing. This idea had calculus, opening up the exploration his laws of planetary motion
already been mooted by ancient into maxima and minima. Calculus and his astronomical tables.
A year after his death in 1630,
Greeks such as Archimedes. is the mathematics of continuous
the transit of Mercury was
Infinitesimals bridge the gap change, and maxima and minima observed as he had predicted.
between continuous things and are the turning points, or limits in
things broken into discrete units. any change – the peak and trough
Kepler then used his cylinder of any graph. Key works
method to find the barrel shapes Pierre de Fermat’s analysis
1609 New Astronomy
with the maximum volume. He of maxima and minima, which
1615 New Solid Geometry of
worked with triangles defined by quickly followed Kepler’s, opened
Wine Barrels
the cylinders’ height, diameter, and the way for the development of 1619 Harmonies of the World
a diagonal from top to bottom. He calculus by Isaac Newton and 1621 Epitome of Copernican
investigated how, if the diagonal Gottfried Leibniz later in the Astronomy
was fixed, like the merchant’s rod, 17th century. ■
THE FLY
ON THE CEILING
COORDINATES
146 COORDINATES

I
n geometry (the study of
IN CONTEXT shapes and measurements),
coordinates are employed to
KEY FIGURE
define a single point – an exact
René Descartes (1596–1650)
position – using numbers. Several
FIELD different systems of coordinates are Problems which can be
Geometry in use, but the dominant one is the constructed by means of
Cartesian system, named after circles and straight lines only.
BEFORE Renatus Cartesius, the Latinized René Descartes
2nd century bce Apollonius name of French philosopher René describing geometry
of Perga explores positions of Descartes. Descartes presented
points within lines and curves. his coordinate geometry in La
Géométrie (Geometry, 1637),
c. 1370 French philosopher
one of three appendices to his
Nicole Oresme represents philosophical work Discours de
qualities and quantities as la Méthode (Discourse on the
lines defined by coordinates. Method), in which he proposed interpreted as a series of defined
1591 French mathematician methods for arriving at truth in the points, which changed the
François Viète introduces sciences. The other two appendices way people thought about natural
symbols for variables in were on light and the weather. phenomena. In the case of events
algebraic notation. such as volcanic eruptions or
Building blocks droughts, plotting elements such as
AFTER Coordinate geometry transformed intensity, duration, and frequency
1806 Jean-Robert Argand uses the study of geometry, which had could help to identify trends.
a coordinate plane to represent barely evolved since Euclid had
complex numbers. written Elements in ancient Greece Finding a new method
some 2,000 years earlier. It also There are two accounts of how
1843 Irish mathematician revolutionized algebra by turning Descartes came to develop the
William Hamilton adds two equations into lines (and lines coordinate system. One suggests
new imaginary units, creating into equations). By using Cartesian that the idea dawned on him as
quaternions, which are plotted coordinates, scholars could visualize he watched a fly moving over the
in four-dimensional space. mathematical relationships. Lines, ceiling of his bedroom. He realized
surfaces, and shapes could also be he could plot its position, using

René Descartes The son of a minor noble, René study. In 1649, he was invited by
Descartes was born in Touraine, Christina, Queen of Sweden, to
France, in 1596. His mother died tutor her and to launch a new
shortly after his birth, and he was academy. His weak constitution
sent to live with his grandmother. could not resist the cold winter.
He later attended a Jesuit college, In February 1650, Descartes
then went to study law in Poitiers. caught pneumonia and died.
In 1618, he left France for the
Netherlands and joined the Dutch Key works
States Army as a mercenary.
Around this time, Descartes 1630–33 Le Monde (The World)
began to formulate philosophical 1630–33 L’Homme (Man)
ideas and mathematical theorems. 1637 Discours de la Méthode
Returning to France in 1623, he (Discourse on the Method)
sold his property there in order 1637 La Géométrie (Geometry)
to secure a lifelong income, then 1644 Principia philosophia
moved back to the Netherlands to (Principles of Philosophy)
 THE RENAISSANCE 147
See also: Pythagoras 36–43 ■ Conic sections 68–69 ■ Trigonometry 70–75 ■ Rhumb lines 125 ■ Viviani’s triangle
theorem 166 ■ The complex plane 214–15 ■ Quaternions 234–35

lines are needed. The horizontal line,

called the x-axis, and the vertical
A rectangular ceiling has a length and a width. y-axis are always perpendicular to
each other; the origin is the only
place they will ever meet. The term
for the x-axis is abscissa, while the
y-axis is the ordinate. Two numbers,
one from each axis, “coordinate” to
Two-dimensional coordinates are pinpointed using pinpoint an exact position.
horizontal ( x) and vertical (y) measurements.
When taking a graph reading,
these two numbers are now
presented as a tuple – a strictly
ordered sequence listed inside
brackets. The abscissa (value of x)
The location of a fly on the ceiling can therefore always precedes the ordinate
be expressed in mathematical terms. (value of y) to create the tuple ( x, y).
Although they were conceived
before negative numbers were fully
accepted, coordinates now often
numbers to describe where it The simplest Cartesian coordinate include both negative and positive
was in relation to the two adjacent system is one-dimensional; it values – negative values below and
walls. Another account relates that indicates positions along a straight to the left of the origin; positive
the idea came to him in dreams line. One endpoint of the line is set values above and to the right of
in 1619, when he was serving as as the zero point, and all other the origin. Together, the two axes ❯❯
a mercenary in southern Germany. points on the line are counted from
It was at this time, too, that he is there in equal lengths, or fractions
thought to have figured out the of a length. Just a single coordinate
relationship between geometry number is needed to describe an
and algebra that is the basis of exact point on the line – as when
the coordinate system. measuring a distance with a ruler
from zero to a unit of length. More
commonly, coordinates are used to
describe points on two-dimensional
surfaces that have a length and
width, or within a three-dimensional
space, which also has depth. To
I realized that it was achieve this, more than one number
necessary… to start again line is needed – each starting at
right from the foundations if I the same zero point, or origin. For
wanted to establish anything a point on a plane (a flat two-
in the sciences that was stable dimensional surface), two number
and likely to last.
René Descartes
This edition of La Géometrie (in
Latin because that was the language
of scholars) was printed in 1639.
Descartes originally published the
book in French so it could be read
by less well-educated people.
148 COORDINATES
create a field of points called a and z, ranging from all negative
coordinate plane, which extends values to all positive values, with
outwards in two dimensions with six possible negative and positive
the origin (0,0) at the centre. Any combinations in between.
point on that plane, which could
stretch to infinity, can be described Curved lines Each problem that I solved
exactly using a pair of numbers. La Géométrie sets out what soon became a rule which
became the foundation of the served afterwards to solve
Plotting 3-D space coordinate system. Descartes, other problems.
For a three-dimensional space, the however, was primarily interested René Descartes
coordinates require a third number, in finding out how coordinates
ordered in the tuple ( x, y, z). The z could help him use algebra to
refers to a third axis, which is better understand lines, especially
perpendicular to the plane formed curved lines. In so doing he created
by the x and y axes (see p.151). a new field of mathematics, called
Each pair of axes creates its own analytic geometry, where shapes
coordinate plane; these intersect are described in terms of their way they are constructed using
at right angles to each other, thus coordinates and the relationships a ruler and pair of compasses.
dividing the space into eight zones between a pair of variables, x and y. The ancient method was limiting;
called octants. The coordinates This was very different from Descartes’ new method opened up
within each octant follow one of Euclid’s “synthetic geometry”, in all sorts of new possibilities.
eight sequences of values for x, y, which shapes are defined by the La Géométrie contains much
discussion about curves, which
were the subject of renewed
interest in the 17th century – partly
Any point on a line can be defined using a number x. because treatises by ancient Greek
mathematicians had been newly
translated, but also because curves
featured prominently in fields of
scientific exploration such as
astronomy and mechanics.
Any point on a plane (flat The points in a straight Coordinates make it possible
surface) can be defined line all share the to convert curves and shapes into
using two numbers same relationship algebraic equations, which can
x and y. between x and y. be shown visually. A straight line
that runs diagonally from the origin,

All lines can be described in

All equations can be terms of an algebraic
plotted as a line. equation. With me,
everything turns
into mathematics.
René Descartes

Coordinates allow curves and shapes to be converted into

equations and equations to be plotted.
 THE RENAISSANCE 149
A geometric shape such as the curve of a roller-coaster can be mapped on
to a graph and described in relation to the x and y axes. The straight section
of the curve has the equation y = x.

110
y The highest point of
HEIGHT OF THE ROLLER-COASTER IN METRES

the roller-coaster is
100 100 metres above
the ground and 100
90 metres from the start,
so has coordinates
80 (100,100).

70
60
50
40 The equation
of the roller-
30 coaster's track
in this section
20 is = .
y x
10
x
0,0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250
DISTANCE FROM THE START OF THE ROLLER-COASTER IN METRES

equidistant from both axes, can be The circle equation that central point is (0,0) on an x, y
described using algebra as y = x, In analytic geometry, all circles graph, the circle equation emerges,
and has coordinates (0,0); (1,1); centred on the origin can be by drawing on Pythagoras’s
(2,2), and so on. The line y = 2x defined as r = (x2 + y2), known as theorem. The circle’s radius can be
would follow a steeper path along a the circle equation. This is because conceived as the hypotenuse of a
line including the coordinates (0,0); a circle can be thought of as all the right-angled triangle with short
(1,2); (2,4), for instance. A line points that lie at an equal distance sides x and y, so r2 = x2 + y2, which
running parallel to y = 2x would from a central point (that distance can be rewritten as r = (x2 + y2).
pass through the y axis at a point being the radius of the circle). If The circle can then be plotted on
other than the origin, such as at axes using different values of x and
(0,2). The formula for this particular y that give the same value of r. For
line is y = 2x + 2 and that includes example, if r is 2, then the circle
the points (0,2); (1,4); (2,6).
y crosses the x axis at (2,0) and
Cartesian coordinates help to P(x,y) (2,0), and it crosses the y axis at
reveal the great power of algebra (0,2) and (0,2). All the other points
to generalize relationships. All the r y on the circle can be seen as one ❯❯
straight lines described above
have the same general equation:
y = mx + c, where the coefficient x x Any point P, with coordinates ( x, y),
m is the slope of the line, indicating on the circumference of a circle can be
how much bigger (or smaller) y is connected to the centre of the circle
(0, 0) by a straight line (the circle’s
compared to x. The constant c, radius) that forms the hypotenuse of
meanwhile, shows where the a right-angled triangle with sides
line meets the y axis when x is of length x and y. The equation of
equal to zero. the circle is r 2 = x2 + y2.
150 COORDINATES
Polar coordinates
In mathematics, polar
coordinates, which define
points on a plane using two
numbers, are the closest rivals
to Descartes’ system. The
first number, the radial
coordinate r, is the distance
from the central point – called
the pole, not the origin. The
second number, the angular
coordinate (), is the angle
that is defined as 0° from a
single polar axis. To compare
it with the Cartesian system,
the polar axis would be the
Cartesian x axis, and the polar
coordinates (1,0°) would
replace the Cartesian
corner of a right-angled triangle A modified form of polar coordinates
coordinates (1,0). The polar that gives an aircraft’s destination in
version of the Cartesian moving around in a circle. As the
corner moves around the circle, terms of angle and distance can be
point (0,1) is (1,90°). used as an alternative to GPS.
Polar coordinates are used the short sides of the triangle vary
to help manipulate complex in length, but the hypoteneuse does
numbers plotted on a plane, not because it is always the radius Géométrie into Latin and also
especially for multiplication. of the circle. The line formed by a popularized the use of coordinates
Multiplying complex numbers point moving in this defined way as a mathematical technique.
is simplified when they are is called a locus. This idea was
treated as polar coordinates, developed by the Greek geometer New dimensions
a process that involves Apollonius of Perga about 1,750 Van Schooten and Fermat had
multiplying the radial years before Descartes’ birth. both suggested extending
coordinates and adding the Cartesian coordinates into
angular ones.
Exchange of ideas the third dimension. Today,
In addition to drawing on theorems mathematicians and physicists
Coordinates of A are r,  formulated by the ancient Greeks, use coordinates to go much further
Descartes exchanged ideas with than that and to imagine a space
90° A
other French mathematicians, with any number of dimensions.
135° 45° among them Pierre de Fermat, with Although it is almost impossible
Pole r whom he frequently corresponded. to visualize such a space,
Descartes and Fermat both made mathematicians can use these
180°  0°
use of algebraic notation, the x and tools to describe lines moving
Polar
y system that François Viète had in four, five, or as many spatial
axis introduced at the end of the 16th dimensions as they desire.
century. Fermat also independently Coordinates can also be
225° 315°
developed a coordinate system, but used to examine the relationship
270° he did not publish it. Descartes between two quantities. This
was aware of Fermat’s ideas, no idea was pioneered as long ago
The polar coordinate system doubt using them to improve his as the 1370s, when a French
is often used to calculate the own. Fermat also helped Dutch monk called Nicole Oresme used
movement of objects around,
or in relation to, a central point. mathematician Frans van Schooten rectangular coordinates and the
to understand Descartes’ ideas. geometric forms created by his
Van Schooten translated La results to understand, for instance,
 THE RENAISSANCE 151
z

Coordinates of
A are x, y, z
The triumph of Cartesian
ideas in mathematics… is
in no small degree due to the
Leiden professor Frans
van Schooten.
A
Dirk Struik
Dutch mathematician

O
y

the relationship of elements such

as speed and time, or the links
between heat intensity and the
degree of expansion due to heat. x
Some quantities can be
represented using coordinates 3-D Cartesian coordinates can be used to plot an
known as vectors, and exist in a object that has, for instance, width, depth, and height.
purely mathematical “vector space”. Three axes ( x, y, z) are set at right angles to each other.
Where they meet is the origin (O).
Vectors are quantities with two
values, which can be plotted as a
magnitude (the length of a line) and visualized in this way to make it describes the location of stars in an
a direction. Velocity is a vector as it easier to add and subtract different imaginary sphere centred on Earth
has exactly those values (a quantity sets of values or to manipulate them and extended infinitely into space.
of speed and a direction of motion), in another way. Polar coordinates, determined by
while other vectors, such as Mathematicians in the 19th distance and angles from the centre
Oresme’s heat and expansion, are century also found new purposes of Earth, are also useful for certain
for Cartesian coordinates. They used types of calculation.
them to represent complex numbers Cartesian coordinates remain
(sums of imaginary numbers, such a ubiquitous tool, however, able to
as 1, and real numbers) or plot anything from simple survey
quaternions (the system that data to the movements of atoms.
extends complex numbers) as Without them, breakthroughs
Mathematics is a more vectors plotted in two, three, or such as analytical calculus
powerful instrument of more dimensions. (which divides quantities into
knowledge than any other infinitesimally small amounts)
that has been bequeathed to The key coordinates and advances in space-time and
us by human agency. The Cartesian coordinate system non-Euclidean geometries could
René Descartes is by no means the only one. not have happened. Cartesian
Geographic coordinates plot points coordinates have had an immense
on the globe as angles from preset impact in mathematics, and in
great circles – the Equator and the many fields of science and the arts,
Greenwich Meridian. A similar from engineering and economics to
system, using celestial coordinates, robotics and computer animation. ■
152

A DEVICE OF
MARVELLOUS
INVENTION
THE AREA UNDER A CYCLOID

T
he ancient Greeks puzzled and volumes of curved shapes by
IN CONTEXT over problems relating to slicing them into parallel pieces
areas and volumes of (Cavalieri’s principle, see above
KEY FIGURES
figures bounded by curves. They right), although he did not publish
Bonaventura Cavalieri
compared the areas of shapes his results until six years later. In
(1598–1647), Gilles Personne
by transforming each one into a 1634, Gilles Personne de Roberval
de Roberval (1602–75) square with the same area as the used this method to work out that
FIELD original shape, then compared the the area beneath a cycloid (the arc
Applied geometry sizes of the squares. This was easy traced by the rim of a rolling wheel)
for shapes with straight edges, but is three times the area of the circle
BEFORE curvilinear shapes caused problems. used to generate the cycloid.
c. 240 bce Archimedes These problems remained
investigates the volume unresolved until 1629, when Italian Squaring the circle
and surface area of spheres mathematician and Jesuit priest The ancient Greek mathematician
in his Method Concerning Bonaventura Cavalieri found a Archimedes had used an ingenious
Mechanical Theorems. method for calculating the areas method of exhaustion to determine
1503 French mathematician
Charles de Bovelles gives the This wheel has rolled over a piece of gum. The graph shows the path
first description of a cycloid in of the gum as the wheel rotates, creating a cycloid shape, which, as
de Roberval discovered, has an area three times that of the wheel.
Introductio in geometriam
(Introduction to Geometry).
Path of Direction of travel Wheel
AFTER gum
1656 Dutch mathematician
Christiaan Huygens bases
his invention of the pendulum
clock on the curve of a cycloid.
1693 De Roberval’s solution
to the area of a cycloid is
published more than 60 years
after its discovery and 18
years after his death. Gum on wheel rim
 THE RENAISSANCE 153
See also: Euclid’s Elements 52–57 ■ Calculating pi 60–65 ■ Mersenne primes 124 ■ The problem of maxima 142–43
■ Pascal’s triangle 156–61 ■ Huygens’s tautochrone curve 167 ■ Calculus 168–75

Since this shark-fin shape (left) and triangle (right) are

the same height and the same width at equivalent points
along their height, Cavalieri’s principle states that they can
be sliced into parallel pieces that have similar area.

A pretty result
which I had
not noticed before.
René Descartes Shark-fin Triangle
on de Roberval’s method for finding shape
the area under a cycloid

quadrature, a process which its area by cutting up models of a

involves constructing a square of cycloid and a circle, weighing the
the area between a parabola and a an area equal to a circle, he failed. pieces, and comparing the results.
straight line. It entailed inscribing He knew the surface area of the Around 1628, Frenchman Marin
a triangle of known area to fit inside sphere was four times that of a Mersenne challenged his fellow
the parabola, then inscribing ever circle of the same radius, but could mathematicians, including de
smaller triangles in all the gaps that not find a square that would give Roberval, René Descartes, and
remained. By adding together the the surface area. Pierre de Fermat, to find both the
areas of the triangles, Archimedes area under the arch of a cycloid and
obtained a close approximation of New spins on the problem a tangent to a point on the curve.
the area he sought. The straight- The first description of a cycloid was When de Roberval told Descartes of
edge-and-compass methods of his published by Charles de Bovelles in his success, the latter dismissed it
day, however, had their limitations. 1503. Italian polymath Galileo gave as “so small a result”. Descartes, in
When he tried to calculate the the cycloid its name (from the Greek turn, discovered the tangent to a
surface area of a 3-D sphere using for “circular”) and tried to calculate cycloid in 1638, and challenged de
Roberval and Fermat to do the same.
Gilles Personne de Roberval Only Fermat succeeded.
In 1658, English architect
Born in 1602, in a field near the Collège Royale. He lived Christopher Wren calculated the
Roberval in northern France, frugally, but managed to buy length of an arc of a cycloid as four
where his mother was bringing a farm for his extended family times the diameter of the generating
in the harvest, Gilles Personne and leased out plots to generate circle. The same year, Blaise Pascal
de Roberval was tutored in income. He continued to practise calculated the area of any vertical
classics and mathematics by mathematics all his life. In 1669, slice of a cycloid. He also imagined
the local priest. In 1628, he he invented a set of scales rotating these vertical slices about
moved to Paris, where he known as the Roberval balance. a horizontal axis, and worked out
joined Marin Mersenne’s He died in 1675. the surface area and volume of the
circle of intellectuals. discs swept out by this rotation.
In 1632, de Roberval became
Pascal’s use of infinitely small slices
professor of mathematics at Key work
the Collège Gervais, and two of shapes to solve the properties of
years later he won a competition 1693 Traité des Indivisibles cycloids would lead to the “fluxions”
for a highly prestigious post at (Treaty on Indivisibles) introduced by Isaac Newton as he
developed early calculus. ■
154

THREE
DIMENSIONS
MADE BY TWO
PROJECTIVE GEOMETRY

U
nlike traditional Euclidean which that object is viewed. The
IN CONTEXT geometry, where all 2-D 17th-century French mathematician
figures and objects belong Girard Desargues was a founder of
KEY FIGURE
in the same plane, projective such geometry.
Girard Desargues
geometry is concerned with how The idea of perspective had
(1591–1661)
the apparent shape of an object is been addressed two centuries
FIELD altered by the perspective from earlier by Renaissance artists and
Applied geometry
BEFORE Linear perspective and geometry
c. 300 bce Euclid’s Elements
sets down ideas that will later
constitute Euclidean geometry. X Vanishing
Centre of point
c. 200 bce In Conics, perspectivity x
Apollonius describes the Y
properties of conic sections. y
P Z
1435 Italian architect Leon z
Battista Alberti codifies the
principles of perspective in
De Pictura (On Painting).
AFTER Axis of
perspectivity
1685 In Sectiones Conicæ,
French mathematician and
painter Philippe de la Hire These two triangles are in perspective from a
defines the hyperbola, viewpoint called the centre of perspectivity (P).
parabola, and ellipse. Lines connecting the corresponding vertices
of the triangles (X to x; Y to y, and Z to z) will Perspective makes the
1822 French mathematician always meet at P. If XYZ were a real triangular parallel lines on sides of
and engineer Jean-Victor object, it would appear as the triangle xyz when this flat-roofed building
viewed from P. Desargues’ theorem states that appear as though they
Poncelet writes a treatise lines extending from the corresponding sides of will eventually meet.
on projective geometry. each triangle will always meet on a line known This meeting point is
as the axis of perspectivity. called a vanishing point.
 THE RENAISSANCE 155
See also: Pythagoras 36–43 ■ Euclid’s Elements 52–57 ■ Conic sections 68–69 ■ The area under a cycloid 152–53
■Pascal’s triangle 156–61 ■ Non-Euclidean geometries 228–29

preserved; this means that if three b

points XYZ are on a straight line,
with Y between X and Z, then their a
images xyz are also on a straight
line with y between x and z. An c
Good architecture image of any triangle is another
should be a projection triangle. The corresponding sides
of life itself. of each triangle can be extended to
Walter Gropius meet at three points on a line (axis
German architect of perspectivity), and a line from
each vertex to its corresponding C
vertex and beyond will meet at a A B
point (the centre of perspectivity). When six arbitrary points are drawn
Desargues realized that all conic on a circle and connected as shown
sections are equivalent in this way (Ab, aB; Ac, Ca; Cb, cB), a straight line
under projection. A single invariant can be drawn through the points where
lines of the same colour cross. Using
architects. Fillipo Brunelleschi had property, such as collinearity, needs
projection, this is true for an ellipse too.
rediscovered the principles of linear only to be proved for a single case,
perspective known to the ancient rather than tested on each conic.
Greeks and Romans, and explored Pascal’s “mystic hexagram” infinity (such as the Sun). By adding
them in his architectural plans, theorem, for instance, states that these points at infinity to the
sculptures, and paintings. Fellow the intersections of lines connecting Euclidean plane, each pair of lines
architect Leon Battista Alberti used pairs of six points on a conic all lie meets at a point, including parallel
“vanishing points” to create a sense on a straight line. It can be shown lines, which meet at infinity.
of 3-D perspective and wrote about by connecting six points on a circle, The method was developed into
the use of perspective in art. a proof valid for other conics, too. a full geometry by Poncelet in 1822.
Desargues then considered Today, projective geometry is used
From maps to maths what happens as the vertex of the by architects and engineers in
As Western explorers sailed to new projection cone moves further away. CAD technology, and in computer
lands, they needed accurate maps Parallel rays come from a point at animation for films and gaming. ■
depicting the spherical world in
two dimensions. In 1569, Flemish Girard Desargues friends. Some of his pamphlets
cartographer Gerardus Mercator were later expanded into
devised a method now known as Born in 1591, Girard Desargues published papers. He wrote
“cylindrical map projection”. This lived in Lyon all his life. He came on perspective and applied
can be envisaged as the surface from a family of wealthy lawyers mathematics to practical
of the globe transferred onto a who owned several properties, projects, such as designing a
surrounding cylinder. When the including a manor and a small spiral staircase and a new form
cylinder is cut from top to bottom chateau with fine vineyards. of pump. Desargues died in
and rolled out, it becomes a two- Desargues made several visits 1661. His work was rediscovered
dimensional map. to Paris and, through Marin and republished in 1864.
In the 1630s, Desargues began Mersenne, became friends with
investigating which properties Descartes and Pascal. Key works
Desargues worked initially as
were unchanged (invariant) when
a tutor and later as an engineer 1636 Perspective
an image is projected onto a and architect. He was an 1639 Rough Draft of Attaining
surface (perspective mapping). excellent geometer and shared the Outcome of Intersecting a
While its dimensions and angles his ideas with his mathematical Cone with a Plane
may change, collinearity is
SYMMETRY
IS WHAT WE SEE AT A

GLANCE
PASCAL’S TRIANGLE
158 PASCAL’S TRIANGLE
Pascal’s triangle is created Result of addition
IN CONTEXT by adding together two adjacent 1 1
numbers (as shown by the
KEY FIGURE arrows) to give the sum in the
Blaise Pascal (1623–62) next row. Each row begins and 1 1
ends with the number 1.
FIELDS 1 2 1
Probability, number theory
BEFORE 1 3 3 1
975 Indian mathematician
Halayudha gives the first 1 4 6 4 1
surviving description of
numbers in Pascal’s triangle. 1 5 10 10 5 1

c. 1050 In China, Jia Xian,

1 6 15 20 15 6 1
describes the triangle later
known as Yang Hui’s triangle.
1 7 21 35 35 21 7 1
c. 1120 Omar Khayyam
creates an early version of 1 8 28 56 70 56 28 8 1
Pascal’s triangle.
AFTER

M
1713 Jacob Bernoulli’s athematics is often The triangle is most commonly
Ars Conjectandi (The Art about the identification named after French philosopher
of Conjecturing) develops of number patterns, and and mathematician Blaise Pascal,
Pascal’s triangle. one of the most remarkable number who explored it in detail in his
patterns of all is Pascal’s triangle. Treatise on the Arithmetical
1915 Wacław Sierpinski
Pascal’s triangle is an equilateral Triangle in 1653. In Italy, however,
describes the fractal pattern
triangle built from a very simple it is known as Tartaglia’s triangle
of triangles later known as arrangement of numbers in ever- after mathematician Niccolò
Sierpinski triangles. widening rows. Each number Tartaglia, who wrote about it in
is the sum of the two adjacent the 15th century. In fact, the origins
numbers in the row above. Pascal’s of the triangle date back to ancient
triangle can be any size, ranging India in 450 bce (see box, p.160).
from just a few rows in depth to
any number. Probability theory
While it might seem that such Pascal’s contribution to the triangle
a simple rule for arranging numbers was notable because he set out a
There are two types of mind… could only lead to simple patterns, clear framework for exploring its
the mathematical, and… Pascal’s triangle is fertile ground properties. In particular, he used the
the intuitive. The former for several branches of higher triangle to help lay the foundations
arrives at its views slowly, mathematics, including algebra, of probability theory in his
but they are… rigid; the number theory, probability, and correspondence with fellow French
latter is endowed with combinatorics (the mathematics mathematician Pierre de Fermat.
greater flexibility. of counting and arranging). Many Before Pascal, mathematicians
Blaise Pascal important sequences have been such as Luca Pacioli, Gerolamo
found in the triangle, and Cardano, and Tartaglia had written
mathematicians believe that it about how to work out the chances
may reflect some truths about of dice rolling particular numbers
relationships that we have yet to or hands of cards coming out a
understand between numbers. certain way. Their understanding
 THE RENAISSANCE 159
See also: Quadratic equations 28–31 ■ The binomial theorem 100–01 ■ Cubic equations 102–05 ■ The Fibonacci
sequence 106–11 ■ Mersenne primes 124 ■ Probability 162–65 ■ Fractals 306–11

was shaky at best, and it was

Pascal’s work with the triangle
that pulled the strands together. 1 is at the top of
Each row has one
Pascal’s triangle and
more number than the
Dividing stakes at the start and end of
row above it.
Pascal was asked to look into each row.
probability in 1652 by a notorious
French gambler. Antoine Gombaud,
the Chevalier de Méré, wanted to
know how to divide stakes fairly
if a game of chance was suddenly
broken off. If a game would normally
The resulting
end only when one player had won
triangle of numbers Every number is the sum
a certain number of rounds, for could go on of the two above it.
instance, de Méré wanted to know forever.
if the division of the stakes should
reflect how many rounds each
player had won. Pascal combined
the numbers step by step to any particular face when you roll it you choose a number of objects
represent the rounds played. is 1 ⁄6. In other words, it is a question from a particular number of
The natural consequence was an of noting how many ways the event available options.
ever-widening triangle. As Pascal can occur, and dividing this by the
showed, the numbers in the total number of possibilities. While Binomial calculations
triangle count the number of ways this is easy enough for a single As Pascal realized, the answer lay
various occurrences can combine dice, with multiple dice, or 52 in binomials – expressions with
to produce a given result. playing cards, the calculations two terms, such as x + y. Each
The probability of an event is become complicated. However, row of Pascal’s triangle gives
defined as the proportion of times it Pascal found that the triangle the binomial coefficients for a
will happen. A dice has six faces, could be used to find the number particular power. The zeroth row
so the probability of it landing on of possible combinations when (the top of the triangle) is used ❯❯

Blaise Pascal Born in Clermont-Ferrand, scientific unit of pressure is

France, in 1623, Blaise Pascal called the Pascal. In 1661, he
was a mathematics prodigy. As a launched what may have been
teenager, his father took him to the world’s first public transport
Marin Mersenne’s mathematical service in Paris, with linked
salon in Paris. Around the age of five-person coaches. He died
21, Pascal developed a mechanical from unexplained causes in
adding and subtraction machine, 1662, aged just 39.
the first ever marketed. As well
as his mathematical contributions, Key works
Pascal played an important role
in many scientific developments 1653 Traité du triangle
of the 17th century, including arithmétique (Treatise on
explorations of fluids and the the Arithmetical Triangle)
nature of a vacuum, which 1654 Potestatum Numericarum
contributed to the understanding Summa (Sums of Powers of
of the idea of air pressure: the Numbers)
160 PASCAL’S TRIANGLE
The Bat Country, a jungle gym project
by American artist Gwen Fisher, is a
Sierpinski tetrahedron featuring softball
bats and balls. This tetrahedron is a 3-D
structure made of Sierpinski triangles.

children (the total number of

objects), the probability of one
girl and two boys is 3/8 (the sum
of all the coefficients in the third
row of the triangle is 8, and there
are three ways of having one girl
in a family of three children).
Pascal’s triangle made it simple
to find probabilities. As Pascal’s
triangle can continue forever, this
works with any powers. The
relationship between binomial
coefficients and the numbers
in Pascal’s triangle reveals a
fundamental truth about numbers
and probability.
for the binomial to the power of 0: the coefficients continue to match a
( x + y)0 = 1. For the binomial to the corresponding line on the triangle. Visual patterns
power of 1, ( x + y)1 = 1x + 1y, so For example, in the binomial Pascal’s simple number pattern
the coefficients (1 and 1) correspond ( x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3, proved to be the launchpad, with
to the first row of the triangle (the the coefficients match the third row Fermat’s work, for the mathematics
zeroth row is not counted as a row). of the triangle. The probabilities are of probability, but its relevance
The binomial ( x + y)2 = 1x2 + 2xy calculated by dividing the number does not stop there. For a start, it
+ 1y2 has the coefficients 1, 2, and of possibilities by the total of all the provides a quick way of multiplying
1, as on the second row of Pascal’s coefficients in the row that reflects out binomial expressions to high
triangle. As binomial expansion the total number of objects: for powers, which would otherwise
leads to ever longer expressions, example, in a family of three take a very long time.

The ancient triangle

Mathematicians knew about the 1303 book by Zhu Shijie
Pascal’s triangle long before the entitled Precious Mirror of the
17th century. In Iran, it is known Four Elements.
as Khayyam’s triangle after Omar The most ancient references
Khayyam, but he was just one to Pascal’s triangle, however,
of many Islamic mathematicians come from India. It appears in
to have studied it between the 7th Indian texts from 450 bce as a
and 13th century – a golden age guide to poetic metre, by the
for learning. In China, too, c. 1050, name of “The Staircase of Mount
Jia Xian created a similar triangle Meru”. The mathematicians of
to show coefficients. His triangle ancient India also realized that
Myanmar’s Hsinbyume pagoda was taken up and popularized by the shallow diagonal lines of
represents the mythical Mount Meru, Yang Hui in the 1200s, which is numbers in the triangle showed
whose staircase inspired another why it is known in China as Yang what are now known as
name for Pascal’s triangle. Hui’s triangle. It is illustrated in Fibonacci numbers (see right).
 THE RENAISSANCE 161
Mathematicians are continually origami, where a Sierpinksi triangle
finding new surprises in it. Some of is converted into three dimensions
the patterns in Pascal’s triangle are to create a Sierpinski tetrahedron.
extremely simple. The outside edge
is entirely made up of the number 1, Number theory
and the next set of numbers, in the I cannot judge my work while There are also many more complex
first diagonal, is a simple number I am doing it. I have to do as patterns hidden within the triangle.
line of 1, 2, 3, 4, 5, and so on. painters do, stand back and One of the patterns found in
One particularly appealing view it from a distance, but Pascal’s triangle is the Fibonacci
property of Pascal’s triangle is the not too great a distance. sequence, which lies on a shallow
“hockey stick” pattern, which can Blaise Pascal diagonal (see below). Another link
be used for addition. If you take a to number theory is the discovery
diagonal down from any of the that the sum of all the numbers
outer 1s, then stop anywhere, in the rows above a given row is
you can then find the total sum of always one less than the sum of the
all of the numbers in the diagonal numbers in the given row. When
by taking one step further in the the sum of all the numbers above
opposite direction. For example, creates a pattern of triangles a given row is a prime, it is a
starting at the fourth 1 on the left identified by Polish mathematician Mersenne prime – a prime number
edge and going down diagonally Wacław Sierpinski in 1915. This that is one less than a power of 2,
to the right, if you stop at the pattern can be made without such as 3 (22 - 1), 7 (23 - 1), and 31
number 10, then the total of the Pascal’s triangle by breaking an (25 - 1). The first list of these
numbers passed so far (1 + 4 + 10) equilateral triangle into ever primes was made by Pascal’s
can be found by going one diagonal smaller triangles by connecting contemporary, Marin Mersenne.
step down to the left: 15. the midpoints of each of the Today, the largest known Mersenne
Colouring in all of the numbers triangles’ three sides. The division prime is 282,589,933 -1. If Pascal’s
divisible by a particular number can continue indefinitely. Today, triangle were drawn at a sufficiently
creates a fractal pattern, while Sierpi nski triangles are popular for large scale, this number would be
colouring all of the even numbers use in knitting patterns and in found there. ■

1 The numbers on the left

1 1 form the Fibonacci sequence,
which can be calculated by
1 adding the numbers on the
1 1 1 shallow diagonals (indicated
1+1 here by the colour shading)
2 1 2 1 of Pascal’s triangle.

1+2
3 1 3 3 1
1+3+1
5 1 4 6 4 1
1+4+3
8 1 5 10 10 5 1

1+5+6+1
13 1 6 15 20 15 6 1

1 + 6 + 10 + 4
21 1 7 21 35 35 21 7 1

1 + 7 + 15 + 10 + 1 1 8 28 56 70 56 28 8 1
34
162
IN CONTEXT

CHANCE IS
KEY FIGURES
Blaise Pascal (1623–62),
Pierre de Fermat (1601–65)

BRIDLED AND
FIELD
Probability
BEFORE

GOVERNED
1620 Galileo publishes Sopra
le Scoperte dei Dadi (On the
Outcomes of Dice), calculating

BY LAW
the chances of certain totals
when throwing dice.
AFTER
1657 Christiaan Huygens
PROBABILITY writes a treatise on probability
theory and its applications to
games of chance.
1718 Abraham de Moivre
publishes The Doctrine
of Chances.
1812 Pierre-Simon Laplace
applies probability theory to
scientific problems in Théorie
analytique des probabilités
(Theory of Probabilities).

B
efore the 16th century,
predicting the outcome of
a future event with any
degree of accuracy was thought
to be impossible. However, in
Renaissance Italy, scholar Gerolamo
Cardano produced in-depth
analyses of outcomes involving
dice. In the 17th century, such
problems attracted the attention
of French mathematicians Blaise
Pascal and Pierre de Fermat. More
renowned for findings such as
Pascal’s triangle (see pp.156–61)
and Fermat’s last theorem (see
pp.320–23), the two men took the
mathematics of probability to a
new level, laying the foundations
for probability theory.
 THE RENAISSANCE 163
See also: The law of large numbers 184–85 ■ Bayes’ theorem 198–99 ■ Buffon’s
needle experiment 202–03 ■ The birth of modern statistics 268–71

frequency, gives the probability

of throwing a six, which can be
expressed as a fraction, a decimal,
or a percentage. This, however, is
an observed finding, based on
Probability theory actual experiments. Theoretical
is nothing but probability of any single event is
common sense reduced calculated by dividing the number
to calculation. of desired outcomes by the total
Pierre-Simon Laplace number of possible outcomes. With
one roll of a six-sided die, the Pierre de Fermat
probability of throwing a six is 1 ⁄6;
the probability of throwing any Born in in 1601 in Beaumont-
other number is 5 ⁄6. de-Lomagne in France, Pierre
de Fermat moved to Orléans
Estimating the odds in 1623 to study law and soon
Forecasting the outcomes of games One popular game in 17th-century began to pursue his interest
of chance proved a useful way of France involved two players taking in mathematics. Like other
approaching probability, which, by turns to throw four dice in a bid to scholars of his day, he studied
definition, measures the likelihood obtain at least one “ace”, or six. The geometry problems from the
of something occurring. For players contributed equal stakes ancient world and applied
example, the chances of throwing a and agreed, in advance, that the algebraic methods to try to
solve them. In 1631, Fermat
six with a die can be estimated by first one to win a certain number
moved to Toulouse and
throwing the die a given number of rounds would take the whole
worked as a lawyer.
of times and dividing the amount of stake. Writer and amateur In his spare time, Fermat
sixes thrown by the total number mathematician Antoine Gombaud, continued his mathematical
of throws. The result, called relative who styled himself Chevalier ❯❯ investigations, circulating his
ideas in letters to friends, such
as Blaise Pascal. In 1653, he
Impossible Unlikely Evens Likely Certain was struck down by plague
but survived to do some of
0 0.5 1
his best work. As well as his
A blue sweet will be A blue sweet will be pulled A blue sweet will be ideas on probability, Fermat
pulled from a jar filled from a jar filled evenly with pulled from a jar filled pioneered differential calculus,
only with pink sweets. both blue and pink sweets. only with blue sweets. but is best remembered for his
contribution to number theory
and Fermat’s last theorem. He
died in Castres in 1665.

Key works

1629 De tangentibus linearum

curvarum (Tangents of Curves)
1637 Methodus ad
disquirendam maximam
Probability is easily measured in the cases shown here. et minimam (Methods of
It is zero if the element in question (blue sweets) is absent, Investigating Maxima
and 0.5 (or 1 ⁄2, or 50 per cent) if half of all sweets are blue. and Minima)
When events are certain, probability = 1 (or 100 per cent).
164 PROBABILITY
was interrupted before completion.
The likelihood of something occurring can be This was known as the “problem
measured mathematically by… of points”, and it had a long history.
In 1494, Italian mathematician
Luca Pacioli had suggested that
the stakes should be divided in
…conducting experiments proportion to the number of rounds
…dividing the number of
to see how often a already won by each player.
desired outcomes…
single event happens… In the mid-16th century, Niccolò
Tartaglia, another prominent
mathematician, had noted that
such a division would be unfair
if the game was interrupted, say,
…and dividing the result by …by the total number of after only one round. His solution
the number of attempts. possible outcomes. was to base the division of the pot
on the ratio between the size of
the lead and the length of the game,
but this also gave unsatisfactory
results for games with many
This is called This is called rounds. Tartaglia remained unsure
relative frequency. theoretical probability. whether the problem was solvable
in a way that would convince all
players of its fairness.

de Méré, understood the 1 ⁄6 odds of ace from 24 throws of a pair of dice The Pascal–Fermat letters
an ace with one throw of a die, and was less likely than one ace from During the 17th century, it was
sought to calculate the odds of four throws of a single die. common for mathematicians to
throwing a double ace using a In 1654, de Méré consulted his meet at academies – scientific
pair of dice. friend Pascal about this problem, societies. In France, the leading
De Méré suggested that the and about the further question of academy was that of the Abbé
chance of getting two aces from how a stake should be divided Marin Mersenne, a Jesuit priest
two throws of a dice was 1 ⁄36, that between the players when a game and mathematician who held
is, 1 ⁄6 as likely as getting an ace
with one die in one roll. To make
these odds the same, he argued
that a pair of dice should be rolled
six times for each roll of the single
die. To have the same chance of
rolling a double ace as you would
from getting one ace when four
dice are thrown, the pair should be
thrown 6  4  24 times. De Méré
consistently lost the bet and was
compelled to deduce that a double

On a standard roulette wheel,

there is a 1/37 chance of the ball
landing on any given number for
a single spin of the wheel. This
number gets closer to 1 the
greater the number of spins.
 THE RENAISSANCE 165
one ace is (1 - 0.233), or 0.7677. game. If they stop at this point,
The game clearly favours the one each should take back his
who makes the throws, rather than stake of 32 pistoles.
the “banker”. Pascal’s step-by-step methods
and Fermat’s considered replies
Choice means probability, Laying down the theory provide some of the earliest
and probability means In other letters, Pascal and Fermat examples of using expectations
mathematicians can discuss other cases of interrupted when reasoning about probability.
get to work. games, such as when the play The correspondence between the
Hannah Fry alternates between two players two laid down basic principles of
British mathematician until one is successful. Fermat probability theory, and games of
notes that what matters is the chance would continue to prove
number of throws remaining when fertile ground for early theorists.
the game stops. He points out that Dutch physicist and mathematician
a player with a 7–5 lead in a game Christiaan Huygens wrote a treatise
to 10 aces has the same chance of translated as “On reasoning in
eventually winning as a player with games of chance”, which was the
weekly meetings at his Paris home. a 17–15 lead in a game to 20. first book on probability theory.
Pascal attended these meetings, Pascal gives an example with An early version of the law
but he and Fermat had never met. two opponents playing a sequence of large numbers (LLN) – a theorem
Nonetheless, having pondered de of games, each with an equal examining the results of performing
Méré’s problems, Pascal chose to chance of winning, where the first the same action (such as throwing
write to Fermat, communicating to win three games wins the stake. a die) a number of times – was part
his thoughts on these and related Each player has staked 32 pistoles, of Swiss mathematician Jacob
issues and asking for Fermat’s own so the stake is 64 pistoles. Over Bernoulli’s Ars Conjectandi (The
views. This was the first of the the course of three games, the first Art of Conjecturing, 1713). In the
letters between Pascal and Fermat player wins twice and the other late 18th and early 19th century,
in which the mathematical theory once. If they now play a fourth Pierre-Simon Laplace applied
of probability was developed. game and the first player wins, probability theory to practical and
then he will take the 64 pistoles; scientific problems, setting out his
Player versus banker if the other wins it, they will have methods in his Théorie Analytique
The Pascal–Fermat letters were each won two games and are des Probabilités (Analytic Theory of
sent via Pierre de Carcavi, a mutual equally likely to win the final Probabilities) in 1812. ■
friend. Seven letters exchanged in
1654 reveal the two men’s thoughts Probability theory Pascal and Fermat, although
on the points problem, which they their letters on the subject
examined in different scenarios. While ancient and medieval contributed much to
They discuss a game between a law graded probability in the subsequent theory.
player attempting to throw at least assessment of judicial evidence, In the late 1700s, Pierre-
one ace in eight throws and a there was no theory on which to Simon Laplace extended the
“banker” who takes the pot if the base it. Similarly, in Renaissance scope of probability theory
player is unsuccessful. If the game times, when insurance was to science, and introduced
is interrupted before an ace has calculated for ships, premiums his mathematical tools for
been thrown, Pascal seems to were based on an intuitive predicting the probability of
suggest that the stakes should be estimate of risk. Odds were a many incidents, including
feature of gaming, but Gerolamo natural phenomena. He also
allocated according to the players’
Cardano was the first to apply recognized its application in
expectations of winning. At the
mathematics to the study of statistics. Probability theory is
start of the game, the probability probability. Games of chance also used in many other fields,
of eight rolls of the die without were the focus of such studies such as psychology, economics,
success is (5 ⁄6)8  0.233, and the even after the deaths of both engineering, and sport.
probability of throwing at least
166

THE SUM OF THE

DISTANCE EQUALS
THE ALTITUDE
VIVIANI’S TRIANGLE THEOREM

I
talian mathematician
IN CONTEXT Vincenzo Viviani studied
under Galileo in Florence. p
KEY FIGURE
After Galileo’s death in 1642,

ALTITUDE
Vincenzo Viviani
Viviani gathered together his
(1622–1703) p
master’s work, editing the first q q
FIELD collected edition in 1655–56.

h
Geometry Viviani’s research included r
work on the speed of sound, which r
BEFORE he measured to within 25 m (82 ft)
c. 300 bce Euclid defines a per second of its true value. He is BASE a
triangle in his book Elements best known, however, for his
and proves many theorems triangle theorem, which states that The altitude in an equilateral
concerning triangles. the sum of the distances between triangle, such as the above, is always
any point in an equilateral triangle equal to the combined length of lines
c. 50 ce Heron of Alexandria and that triangle’s sides is equal to
drawn from any point in the triangle
defines a formula for finding perpendicular to its three sides.
the altitude (height) of the triangle.
the area of a triangle from
its side lengths. Proving the theorem The area of a triangle is 1/2  base
AFTER Starting with an equilateral  height, so if the lengths of the
1822 German geometer Karl triangle of base (side) a, and an perpendiculars are p, q, and r,
Wilhelm Feuerbach publishes altitude of h (see above right), a the areas of the triangles add up
point is made inside the triangle. to 1/2 (p + q + r)a. This is also the
a proof for the nine-point ircle,
Perpendicular lines (p, q, and r ) area of the large triangle, which
which passes through the
are drawn from that point to each is 1/2 ha, and so h = p + q + r.
midpoint of each side of of the three sides, meeting each If you were to break a stick of
a triangle. side at 90°. The triangle is divided length h into three, there would
1826 Swiss geometer Jakob into three smaller triangles by always be a point in the triangle
Steiner describes the triangle drawing a line from the point to from which the pieces form the
centre that has the minimum each corner of the main triangle. perpendiculars p, q, and r. ■
sum of distances from the
triangle's three vertices. See also: Pythagoras 36–43 ■ Euclid’s Elements 52–57 ■ Trigonometry 70–75
■ Projective geometry 154–55 ■ Non-Euclidean geometries 228–29
 THE RENAISSANCE 167

THE SWING OF
A PENDULUM
HUYGENS’S TAUTOCHRONE CURVE

I
n 1656, Dutch physicist and shaped “cheeks”. In theory, the time
IN CONTEXT mathematician Christiaan of each movement would now be
Huygens created the pendulum the same from any starting point.
KEY FIGURE
clock, a clock with a swinging However, friction introduced a
Christiaan Huygens
weight that was constant. He larger error than the one Huygens
(1629–95)
wanted to resolve the navigational was trying to resolve. It was only in
FIELD problem of determining a ship’s the 1750s that the Italian Joseph-
Geometry longitude. This was impossible Louis Lagrange arrived at a
without precise calculations of solution, where the height of the
BEFORE time, so it required an accurate curve needs to be in proportion to
1503 French mathematician clock to cope with the rolling the square of the length of the arc
Charles de Bovelles is the first motion of the waves, which caused travelled by the pendulum. ■
to describe a cycloid. wide variations in pendulum swing,
1602 Galileo discovers that leading to time discrepancies.
the time taken for a pendulum
Seeking the right curve
to complete a swing does not
The key lay in finding a curved
depend on the swing’s width.
path for the pendulum to follow I was… struck by the
AFTER (known as a tautochrone curve), remarkable fact that in
1690 Swiss mathematician whereby the time the pendulum geometry all bodies gliding
Jacob Bernoulli draws on takes to return to its lowest point along the cycloid… descend
Huygens’s imperfect solution is constant whatever its highest
from any point in precisely
to the tautochrone problem to point. Huygens identified the
cycloid, a curve that was steep
the same time.
solve the brachistochrone Herman Melville
problem – finding a curve at the top and shallow at the Moby Dick (1851)
bottom. The curved path of any
of the fastest descent.
pendulum would have to be
Early 1700s The longitude adjusted so it travelled in a cycloid.
problem is resolved by British Huygens’s idea was to constrain
clockmaker John Harrison and the pendulum by adding cycloid-
others – using springs rather
than pendulums. See also: The area under a cycloid 152–53 ■ Pascal’s triangle 156–61 ■ The law of
large numbers 184–85
WITH CALCULUS
I CAN PREDICT THE
FUTURE
CALCULUS
170 CALCULUS

T
he development of calculus,
IN CONTEXT the branch of mathematics
that deals with how things
KEY FIGURES
change, was one of the most
Isaac Newton (1642–1727),
significant advances in the history
Gottfried Leibniz (1646–1716) of mathematics. Calculus can
Nothing takes place in the
FIELD show how the position of a moving
vehicle changes over time, how the
world whose meaning is
Calculus not that of some maximum
brightness of a light source dims as
BEFORE it moves further away, or how the or minimum.
287–212 bce Archimedes position of a person’s eyes alters Leonhard Euler
uses the method of exhaustion as they follow a moving object.
to calculate areas and It can ascertain where changing
volumes, introducing the phenomena reach a maximum or
concept of infinitesimals. minimum value, and at what rate
they travel between the two.
c. 1630 Pierre de Fermat Alongside rates of change,
uses a new technique for another important aspect of generalized quantities, calculus
finding tangents to curves, calculus is summation (the process has its own rules, notations, and
locating their maximum and of adding things), which developed applications, and its development
minimum points. from the need to calculate areas. between the 17th and 19th
AFTER Eventually, the study of areas and centuries led to rapid progress
volumes was formalized into what in fields such as engineering
1740 Leonhard Euler applies
became known as integration, and physics.
the ideas of calculus to
while calculating rates of change
synthesize calculus, complex was termed differentiation. Ancient origins
algebra, and trigonometry. By providing a better The ancient Babylonians and
1823 French mathematician understanding of the behaviour Egyptians were particularly
Augustin-Louis Cauchy of phenomena, calculus can be interested in measurement. It was
formalizes the fundamental used to predict and influence their important for them to be able to
theorem of calculus. future state. In much the same way calculate the dimensions of fields
as algebra and arithmetic are tools for growing and irrigating crops
for working with numerical or and to work out the volume of
buildings to store grain. They
developed early notions of area
and volume, although these tended
He approximated the to be in the form of very specific
Archimedes saw area of the circle by
the circle as having examples, such as in the Rhind
placing it in polygons papyrus, where one problem
an infinite number with infinitesimally
of sides. smaller sides. involves the area of a round field
with a diameter of 9 khet (a khet
being an ancient Egyptian unit
of length). The rules laid down in
the Rhind papyrus led ultimately
to what would become known
Ancient Greek more than 3,000 years later as
Division into infinite
thought is at the parts is essential integral calculus.
foundation of to integration (the study The concept of infinity is central
modern calculus. of areas and volumes). to calculus. In ancient Greece,
Zeno’s paradoxes of motion, a set
of philosophical problems devised
 THE RENAISSANCE 171
See also: The Rhind papyrus 32–33 ■ Zeno’s paradoxes of motion 46–47 ■ Calculating pi 60–65 ■ Decimals 132–37 ■ The
problem of maxima 142–43 ■ The area under a cycloid 152–53 ■ Euler’s number 186–91 ■ Euler’s identity 197

by the philosopher Zeno of Elea polygons infinitely smaller. It was As civilizations developed, accurate
in the 5th century bce, posited that thought that their combined area measurement became essential. This
motion was impossible because would eventually converge towards ancient Egyptian tomb painting shows
surveyors using rope to calculate the
there are an infinite number of the true area of the shape. dimensions of a wheat field.
halfway points in any given This so-called “method of
distance. In around 370 bce, the exhaustion” was taken up by
Greek mathematician Eudoxus Archimedes in around 225 bce. of times for all possible variations,
of Cnidus proposed a method of He approximated the area of a generalized symbols could be used
calculating the area of a shape by circle by enclosing it within to prove that a case is true for all
filling it with identical polygons of polygons with increasing numbers numbers to infinity.
known area, and then making the of sides. As the number of sides Mathematics had suffered a
increases, the polygons (of known long period of stagnation in
area) more closely resemble the Europe but, as the Renaissance
circle. Taking this idea to the limit, took hold in the 14th century,
Archimedes imagined a polygon renewed interest in the subject
with sides of infinitesimally led to fresh ideas about motion
smaller length. The recognition and the laws governing distance
For by the ultimate velocity is of infinitesimals was a pivotal and speed. French mathematician
meant that, with which the moment in the development of and philosopher Nicole Oresme
body is moved, neither before calculus: previously insoluble studied the velocity of an
it arrives at its last place, when puzzles, such as Zeno’s paradoxes accelerating object against time,
the motion ceases nor after of motion, could now be solved. and he realized that the area under
but at the very instant a graph depicting this relationship
when it arrives. Fresh ideas was equivalent to the distance
Isaac Newton Mathematicians in medieval China travelled by the object. This notion
and India made further advances in would be formalized in the late
dealing with infinite sums. In the 17th century by Isaac Newton and
Islamic world, too, the development Isaac Barrow in England, Gottfried
of algebra meant that, rather than Leibniz in Germany, and Scottish
spelling out a calculation millions mathematician James Gregory. ❯❯
172 CALCULUS
Oresme’s work was inspired by solving physical and philosophical French mathematician François
that of the “Oxford Calculators”, problems using calculations and Viète promoted the use of symbols
a 14th-century group of scholars logic, and were interested in the in algebra (which had previously
based at Merton College, Oxford, quantitive analysis of phenomena been described in words), while
who developed the mean speed such as heat, colour, light, and Flemish mathematician Simon
theorem, which Oresme later velocity. They were inspired by the Stevin initiated the concept of
proved. It states that if one body trigonometry of Arab astronomer mathematical limits, whereby
is moving with a uniformly al-Battani (858–929 ce) and the logic the sum of amounts could converge
accelerated motion and a second and physics of Aristotle. to a limiting value, much like the
body is moving with a uniform area of Archimedes’ polygons
speed equal to the mean speed of New developments converged to the area of a circle.
the first body, and both bodies are The incremental steps towards At much the same time,
moving for the same duration, they the development of calculus German mathematician and
will cover the same distance. The gathered pace towards the end of astronomer Johannes Kepler was
Merton scholars were devoted to the 16th century. In around 1600, researching the motion of the
planets, including calculating the
area enclosed by a planetary orbit,
which he recognized as elliptical
rather than circular. Using ancient
Greek methods, he worked out the
area by dividing the ellipse into
strips of infinitesimal width.
A forerunner of the more formal
integration to come, Kepler’s
method was further developed in
1635 by Italian mathematician
Bonaventura Cavalieri in Geometria
indivisibilibus continuorum nova
quadam ratione promota (Geometry,
Advanced in a New Way by the
Indivisibles of the Continua).
Cavalieri worked out a “method
of indivisibles”, which was a more
rigorous method of determining the
size of shapes. More developments
followed in the 17th century with
the work of English theologian and
mathematician Isaac Barrow and
Italian physicist Evangelista
Torricelli, followed by that of Pierre
de Fermat and René Descartes,
whose analysis of curves advanced
the new area of graphical algebra.

This illustration of Kepler’s

Platonic solid model of the Solar
System appeared in a book published
in 1596. Kepler used infinitesimally
small strips to measure the distance
covered in an orbit. This method was
the forerunner of integration.
 THE RENAISSANCE 173
The fundamental
There is an inverse relationship between theorem of calculus
differentiation and integration.
The study of calculus is
underpinned by the
fundamental theorem of
calculus, specifying the
relationship between
Differentiation is differentiation and integration,
used to calculate the Integration is used to both of which rely on the
derivative, which gives the calculate the integral, which concept of infinitesimals. First
gradient to a curve; this is gives the area bounded articulated by James Gregory
the rate of change at any by a curve. in his 1668 Geometriae Pars
given point. Universalis (The Universal Part
of Geometry), it was then
generalized by Isaac Barrow
in 1670, and formalized in 1823
by Augustin-Louis Cauchy.
The theorem has two
parts. The first states that
The derivative can be The integral can be integration and differentiation
used to calculate the speed used to determine the are opposites – for any
at which a falling object is area of a 2-D shape or continuous function (one
travelling at any given the volume of a that can be defined for all
moment in time. 3-D shape. values), there exists an “anti-
derivative” (or “integral”),
whose derivative (a measure
of the rate of change) is the
Fermat also located maxima and calculus (or differentiation), function itself. The second part
minima, the greatest and least which together with the related of the theorem states that if
values of a curve. field of integral calculus led to the values are inserted into the
fundamental theorem of calculus anti-derivative F ( x), the
Fluxion model (see box, right). The idea of result – the definite integral
In 1665–66, English mathematician differential calculus is that the rate of the function f (x) – makes
Isaac Newton developed his at which a variable changes at a it possible to calculate
areas under the curve of
“method of fluxions”, a method for point is equal to the gradient of a
the function f (x).
calculating variables that changed tangent at that point. This can be
over time, which was a milestone pictured by drawing a tangent ❯❯
in the history of calculus. Like
Kepler and Galileo, Newton was
interested in studying moving
y
bodies and was particularly
keen to unify the laws governing
the motion of celestial bodies
with motion on Earth.
In Newton’s fluxion model, he x
considered a point moving along
0 1 2 3
a curve as being divided into two
perpendicular components (x and y),
Differentiation can be used to find the
and then considered the velocities rate of change at a given point in time. James Gregory (1638–75) was
of those components. This work The blue line shows the rate of change the first person to formulate the
laid the foundation for what overall and the orange tangent shows fundamental theorem of calculus.
became known as differential the rate of change at a given point.
174 CALCULUS
(a straight line that touches a curve Isaac Newton’s Opticks, a treatise
at only one point). The gradient or about the reflections and refractions
steepness of this line will be the of light, published in 1704, contains
the first details of his work in the
rate of change of the curve at area of calculus.
that point. Newton recognized that
at the maxima and minima, the
gradient of the curve was zero, Significant contributions to
because when something is at calculus were also made in
its highest or lowest point, it is Switzerland by the brothers
momentarily not changing. Newton Jacob and Johann Bernoulli, who
went on to develop his theory coined the term “integral” in 1690.
further by considering the converse Scottish mathematician Colin
problem – if the rate at which a Maclaurin published his Treatise
variable changes is known, is it on Fluxions in 1742, promoting and
possible to calculate the shape furthering Newton’s methods,
of the variable itself? This “anti- and attempting to make them
differentiation” entailed working more rigorous. In this work,
out areas under the curve. Maclaurin applies calculus to
the study of infinite series of
Newton v. Leibniz algebraic terms. Meanwhile
Around the time that Newton was Swiss mathematician Leonhard
developing his calculus, German this time about integration, again Euler, a close friend of Johann
mathematician Gottfried Leibniz using different notation from that Bernoulli’s sons, was influenced
was working on his own version, of Newton. In an unpublished by their ideas on the subject. In
based on the consideration of manuscript dated 29 October 1675, particular, he applied the idea of
infinitesimal changes in the two Leibniz was the first person to infinitesimals to what is known
x
coordinates defining a point on a use the “integral” symbol ∫, as the exponential function, e .
curve. Leibniz used very different which is used and recognized This ultimately led to “Euler’s
identity”, e + 1 = 0, an equation
i
notation from Newton’s, and in universally today.
1684 published a paper on what There was much debate about that connects five of the most
would later become known as who discovered modern calculus fundamental mathematical
differential calculus. Two years first: Newton or Leibniz. It led to quantities (e, i, , 0, and 1) in
later, he published another paper, protracted bitterness between the a very simple way.
two rivals and across much of
the mathematical community.
Although Newton devised his
theory of fluxions in 1665–66, he
did not publish it until 1704, when
Assuming I know our it was added as an appendix to
his work Opticks. Leibniz began When the values successively
instantaneous speed at every assigned to the same variable
possible moment, can I then to devise his version of calculus
around 1673, and published it in indefinitely approach a fixed
use that information to value, so as to end by differing
1684. Newton’s subsequent
determine how far we’ve from it as little as desired, this
Principia is said by some to have
travelled? Calculus says I can. been influenced by Leibniz’s work. fixed value is called the limit.
Jennifer Ouellette By 1712, Leibniz and Newton Augustin-Louis Cauchy
American science writer
were openly accusing one another
of plagiarism. The modern
consensus is that Leibniz and
Newton developed their ideas
on the subject independently.
 THE RENAISSANCE 175
The notation of modern calculus

 Invented by Newton for differentiation.

 Invented by Leibniz for integration.

dy/dx Invented by Leibniz for differentiation.

 Invented by Lagrange for differentiation.

As the 18th century progressed, criteria for which functions would

calculus proved increasingly be integrable or not, based on
Gottfried Leibniz
useful as a tool for describing and defining finite upper and lower
understanding the physical world. limits for the function. Born in Leipzig, Germany, in
In the 1750s, Euler, working in 1646, Gottfried Leibniz was
collaboration with French Ubiquitous applications raised in an academic family.
mathematician Joseph-Louis Many advances in physics His father was a professor
Lagrange, used calculus to provide and engineering have relied on of moral philosophy, while
an equation – the Euler–Lagrange calculus. Albert Einstein used his mother was the daughter
equation – for understanding both it in his theories of special and of a professor of law. In
fluid (gas and liquid) and solid general relativity in the early 1667, after completing his
mechanics. In the early 19th 20th century, and it has been university studies, Leibniz
century, French physicist and applied extensively in quantum became an advisor on law
mathematician Pierre-Simon mechanics (dealing with the and politics to the Elector of
Mainz, a role that enabled
Laplace developed electromagnetic motion of subatomic particles).
him to travel and meet other
theory with the help of calculus. Schrödinger’s wave equation, European scholars. After
a differential equation published his employer’s death in 1673,
Formalizing the theories in 1925 by Austrian physicist he took up the role of librarian
The various developments in Erwin Schrödinger, models a to the Duke of Brunswick
calculus were formalized in 1823 particle as a wave whose state in Hanover.
when Augustin-Louis Cauchy can only be determined by Leibniz was a celebrated
formally stated the fundamental using probability. This was philosopher as well as a
theorem of calculus. In essence, groundbreaking in a scientific mathematician. He never
this states that the process of world that had up until then been married and died in 1716 to
differentiation (working out rates of governed by certainty. little fanfare. His successes
change of a variable represented by Calculus has many important had been overshadowed by his
a curve) is the inverse of the process applications today; it is used, for calculus dispute with Newton
of integration (calculating the area instance, in search engines, and were only recognized
several years after his death.
beneath a curve). Cauchy’s construction projects, medical
formalization allowed calculus to advances, economic models, and
be regarded as a unified whole, weather forecasts. It is difficult Key works
dealing with infinitesimals in a to imagine a world without this
1666 On the Art of
consistent way using universally all-pervasive branch of mathematics,
Combination
agreed notation. as it would most certainly be one
1684 New Method for
The field of calculus was further without computers. Many would Maximums and Minimums
developed later in the 19th century. argue that calculus is the most 1703 Explanation of Binary
In 1854, German mathematician important mathematical discovery Arithmetic
Bernhard Riemann formulated in the last 400 years. ■
176

THE PERFECTION
OF THE SCIENCE OF
NUMBERS
BINARY NUMBERS

I
n everyday life we are used to system is a base-2 counting system
IN CONTEXT the base-10 counting system and employs just two symbols,
with its familiar ten digits, 0 to 0 and 1. Rather than increasing
KEY FIGURE
9. When we count from 10 onwards, in multiples of 10, each column
Gottfried Leibniz (1646–1716)
we put a 1 in the “tens” column and represents a power of 2. So the
FIELDS a 0 in the “units” column, and so binary number 1011 is not 1,011
Number theory, logic on, adding columns for hundreds, but 11 (from right to left: one 1,
thousands, and beyond. The binary one 2, no 4s, and one 8).
BEFORE
c. 2000 bce Ancient Egyptians
use a binary system of Decimal Binary number Binary visual
doubling and halving to carry numbers 16s 8s 4s 2s 1s 16s 8s 4s 2s 1s
out multiplication and division.
1 0 0 0 0 1
c. 1600 English mathematician
and astrologer Thomas Harriot 2 0 0 0 1 0
experiments with number 3 0 0 0 1 1
systems, including binary.
4 0 0 1 0 0
AFTER
1854 George Boole uses 5 0 0 1 0 1
binary arithmetic to develop
Boolean algebra. 6 0 0 1 1 0

1937 Claude Shannon shows 7 0 0 1 1 1

how Boolean algebra could be
implemented using electronic 8 0 1 0 0 0
circuits and binary code. 9 0 1 0 0 1
1990 A 16-bit binary code
10 0 1 0 1 0
is used to code pixels on a
computer screen, allowing
Binary numbers are written as 1s and 0s, using a base-2 system.
it to display more than This chart shows how to write the numbers 1 to 10, from the base-10
65,000 colours. system, as both binary numbers and binary visuals – which is how
a computer would process them – where 1 is “on” and 0 is “off”.
 THE RENAISSANCE 177
See also: Positional numbers 22–27 ■ The Rhind papyrus 32–33 ■ Decimals 132–37 ■ Logarithms 138–41
■ The mechanical computer 222–25 ■ Boolean algebra 242–47 ■ The Turing machine 284–89 ■ Cryptography 314–17

Bacon’s cipher be any two different objects –

“… as by bells, by trumpets, by
English philosopher and courtier lights and torches… and any
Francis Bacon (1561–1626) was instruments of like nature”. It
Reckoning by twos, a great dabbler in cryptography, was an ingeniously adaptable
that is, by 0 and 1… is the or the science of deciphering cipher, which Bacon could use
most fundamental way of codes. He developed what he to “make anything signify
called a “biliteral” cipher, which anything”. A secret message
reckoning for science, and used the letters a and b to could be hidden in a group of
offers up new discoveries, generate the entire alphabet – objects or numbers, or even
which are… useful, even for a = aaaaa, b = aaaab, c = aaaba, musical notation. Samuel
the practice of numbers. d = aaabb, and so on. If you Morse’s dot–dash telegraph
Gottfried Leibniz substitute 0 for a and 1 for b, code, which revolutionized
this becomes a binary sequence. communication in the 19th
It is an easy code to break, but century, and the on/off encoding
Bacon realized that a and b do in a modern computer both have
not have to be letters – they can parallels with Bacon’s cipher.

Binary choices are black and white; The potential of binary numbers Leibniz saw links between this
in any column there is only ever 1 was finally realized by German binary approach to divination and
or 0. This simple “on or off” concept mathematician and philosopher his work with binary numbers.
has proved vital in computing, for Gottfried Leibniz. In 1679, he Above all, Leibniz was driven
example, where every number can described a calculating machine by his religious faith. He wanted
be represented by a series of that worked on binary principles, to use logic to answer questions
switch-like on/off actions. with open or closed gates to let about God’s existence and believed
marbles fall through. Computers that the binary system captured his
Binary power revealed work in a similar way, using view of the Universe’s creation,
In 1617, Scottish mathematician switches and electricity rather with 0 representing nothingness
John Napier announced a binary than gates and marbles. and 1 representing God. ■
calculator based on a chessboard. Leibniz outlined his ideas
Each square had a value, and that on the binary system in 1703 in
square’s value was “on” or “off” Explanation of Binary Arithmetic,
depending on whether a counter showing how 0s and 1s could
was placed on the square. The represent numbers and so simplify
calculator could multiply, divide, even the most complex operations
and even find square roots, but into a basic binary form. He had
was considered a mere curiosity. been influenced by correspondence
Around the same time, Thomas with missionaries in China, who
Harriot was experimenting with introduced him to the I Ching, an
number systems, including the ancient Chinese book of divination.
binary system. He was able to The book divided reality into the
convert base-10 numbers to binary two opposing poles of yin and
and back again, and could also yang – one represented as a broken
The teaching and commentaries
calculate using binary numbers. line, the other as an unbroken line. on the I Ching of ancient Chinese
However, Harriot’s ideas remained These lines were displayed as philosopher Confucius (551–479 bce)
unpublished until long after his six-line hexagrams, combined into influenced the work of Leibniz and
death in 1621. a total of 64 different patterns. other 17th–18th-century scientists.
THE
ENLIGHT
1680–1800
ENMENT
180 INTRODUCTION

Jacob Bernoulli discovers In the posthumously Buffon’s needle

an approximate value of the published Ars Conjectandi experiment Abraham de Moivre
irrational number e while (The Art of Conjecturing), demonstrates the publishes his paper
exploring compound Jacob Bernoulli explains the link between detailing normal
interest on loans. law of large numbers. probability and pi. distribution.

1683 1713 1733 1738

1687 1727 1736

The three laws of The constant e, one of the Euler’s attempt to solve the old
motion are summarized most important values problem of the Königsberg
by Isaac Newton in his in maths, is given Bridges leads to graph theory
Principia Mathematica. its notation by and important developments in
Leonhard Euler. mathematical topology.

B
y the late 17th century, brothers Jacob and Johann Johann’s sons, and at an early age
Europe had become Bernoulli further developed the proved himself a worthy successor
established as the cultural theory of calculus in their “calculus to Jacob and Johann. Aged only 20,
and scientific centre of the world. of variations” and several other he suggested a notation for the
The Scientific Revolution was well mathematical concepts discovered irrational number e, for which
under way, inspiring a new, rational in the 17th century. The elder Jacob Bernoulli had calculated
approach not only to the sciences, brother, Jacob, is recognized for an approximate value.
but to all aspects of culture and his work on number theory, but Euler published numerous
society. The Age of Enlightenment, he also helped develop probability books and treatises, and worked
as this period came to be known, theory, introducing the law of in every field of mathematics, often
was a time of significant socio- large numbers. recognizing the links between
political change, and produced an Along with their mathematically apparently separate concepts of
enormous increase in the spread gifted children, the Bernoullis were algebra, geometry, and number
of knowledge and education during the leading mathematicians of the theory, which were to become
the 18th century. It was also a early 18th century, making their the basis for further fields of
period of considerable progress home town of Basel in Switzerland Mathematical study. For example,
in mathematics. a centre of mathematical study. his approach to the seemingly
It was here that Leonhard Euler, simple problem of planning a route
Swiss giants the next, and arguably greatest, through the city of Königsberg,
Building on the work of Newton Enlightenment mathematician, crossing each of its seven bridges
and Leibniz, whose ideas were was born and educated. Euler was only once, uncovered much deeper
finding practical application in a contemporary and friend concepts of topology, inspiring
physics and engineering, the of Daniel and Nicholas Bernoulli, new areas of research.
 THE ENLIGHTENMENT 181

Euler uses the constant

named after him to Joseph-Louis Lagrange
formulate one of the most succeeds Euler as director Thomas Malthus predicts
recognizable equations of mathematics at the a catastrophe resulting
in mathematics, known as Prussian Academy from exponential
“Euler’s identity”. of Sciences. population growth.

1747 1766 1798

1742 1763 1771 1799

The famous conjecture that Bayes’ theorem, which An algebraic Carl Friedrich Gauss
every even integer predicts the likelihood resolution for the roots produces his
greater than 2 is the sum of future events based of polynomials is fundamental theorem
of two primes is proposed on prior knowledge, formulated by Lagrange. of algebra aged 21.
by Christian Goldbach. is established.

Euler’s contributions to all fields Leibniz and the Bernoullis, and In the latter part of the 18th
of mathematics, but in particular corresponded regularly with them century, France became the
calculus, graph theory, and number about their theories. In a letter to European centre of mathematical
theory, were enormous, and he was Euler, he proposed the conjecture enquiry, with Joseph-Louis
also influential in standardizing for which he is best known, that Lagrange in particular emerging
mathematical notation. He is every even integer greater than 2 as a significant figure. Lagrange
especially remembered for the can be expressed as the sum of two had made his name working with
elegant equation known as “Euler’s primes, which remains unproven Euler, but later made important
identity”, which highlights the to this day. contributions to polynomials
connection between fundamental Others contributed to the and number theory.
mathematical constants such development of the growing field
as e and p. of probability theory. Georges-Louis New frontiers
Leclerc, Comte de Buffon, for As the century drew to a close,
Other mathematicians example, applied the principles Europe was reeling from political
The Bernoullis and Euler tended of calculus to probability, and revolutions that had toppled the
to eclipse the achievements of the demonstrated the link between monarchy in France and given birth
many other mathematicians of pi and probability, while another to the United States of America.
the 18th century. Among them Frenchman, Abraham de Moivre A young German, Carl Friedrich
was Christian Goldbach, a German described the concept of normal Gauss, published his fundamental
contemporary of Euler’s. In the distribution, and Englishman theorem of algebra, marking the
course of his career, Goldbach Thomas Bayes proposed a theorem beginning of a spectacular career
had befriended other influential of the probability of events based and a new period in the history
mathematicians, including on knowledge of the past. of mathematics. ■
182

TO EVERY ACTION
THERE IS AN EQUAL
AND OPPOSITE
REACTION
NEWTON’S LAWS OF MOTION

I
n using mathematics to about the force needed to enable
IN CONTEXT explain the movement of a body to move in a circular path.
the planets and of objects on He used his knowledge of forces
KEY FIGURE
Earth, Isaac Newton fundamentally and German astronomer Johannes
Isaac Newton (1642–1727)
changed the way we see the Kepler’s laws of planetary motion
FIELD Universe. He published his findings to deduce how elliptical orbits were
Applied mathematics in 1687 in the three-volume related to the laws of gravitational
Philosophiae Naturalis Principia attraction. In 1686, English
BEFORE Mathematica (Mathematical astronomer Edmond Halley
c.330 bce Aristotle believes it Principles of Natural Philosophy), persuaded Newton to write up his
takes force to maintain motion. often called the Principia for short. new physics and its applications
to planetary motion.
c.1630 Galileo Galilei
How the planets move In his Principia, Newton used
conducts experiments on
By 1667, Newton had already mathematics to show that the
motion and finds that friction developed early versions of his consequences of gravity were
is a retarding force. three laws of motion and knew consistent with what had been
1674 Robert Hooke writes An
attempt to prove the motion of Newton’s second and third law
the Earth and hypothesizes help explain how scales work. When
what will become Newton’s we weigh ourselves, our weight (the
first law. mass of an object multiplied by
gravity) is a force, now measured
AFTER in newtons. Newtons can be
1905 Albert Einstein presents converted into measurements
of mass, such as kilograms.
his theory of relativity, which
challenges Newton’s view of
the force of gravity. The body (mass) of the person on the
scale is pushed down by gravity.
1977 Voyager 1 is launched.
The scale pushes back up with
With no friction or drag in exactly the same force as the
space, the craft keeps going downward pressure from gravity.
due to Newton’s first law, and
The weight is indicated on most
exits the Solar System in 2012. scales in kilograms. One kilogram
is equal to 9.81 newtons.
 THE ENLIGHTENMENT 183
See also: Syllogistic logic 50–51 ■ The problem of maxima 142–43 ■ Calculus 168–75 ■ Emmy Noether and abstract
algebra 280–81

Newton’s three laws of motion

First law: Every body Second law: The change

continues in its state of in motion is proportional
rest, or of uniform motion in to the motive force Third law: To every action
a straight line, unless it is impressed, and is made in there is an equal and
compelled to change that the direction of the straight opposite reaction.
state by forces impressed line in which that force is
upon it. impressed.

observed experimentally. He from the gravitational attraction Newton’s third law says that if two
analysed the motion of bodies between two bodies or an applied objects are in contact, the reaction
under the action of forces and force (such as when a snooker cue forces between them cancel out,
posited gravitational attraction to strikes a ball). The second law each pushing on the other with
explain the movement of the tides, explains what is happening when an equal force but in opposing
projectiles, and pendulums, and objects are in motion. Newton directions. An object resting on
the orbits of planets and comets. said that the rate of change of a table pushes down on it, and the
momentum (mass  velocity) of a table pushes back with an equal
Laws of motion body is equal to the force acting force. If this were not true, the
Newton began Principia by stating on it. If a graph is plotted showing object would move. Until Einstein’s
his three laws of motion. The first velocity against time, then the theory of relativity, the whole of
says that a force is needed to create gradient at any point is the rate of mechanical physics was based on
motion, and that this force may be acceleration (any change in velocity). Newton’s three laws of motion. ■

Isaac Newton Isaac Newton was born on where he developed a rivalry

Christmas Day in 1642 in with eminent English scientist
Lincolnshire, England, and was Robert Hooke. One of several
brought up in early childhood by government positions he held
his grandmother. Newton studied was Master of the Royal Mint,
at Trinity College, Cambridge, where he oversaw the switch
where he showed a fascination of the British currency from the
for science and philosophy. silver to the gold standard. He
During the Great Plague in was also President of the Royal
1665–1666, the university was Society. Newton died in 1727.
forced to close, and it was during
this period that he formulated his Key work
ideas on fluxions (rates of change
at a given point in time). 1687 Philosophiae Naturalis
Newton made significant Principia Mathematica
discoveries in the fields of (Mathematical Principles
gravitation, motion, and optics, of Natural Philosophy)
184

EMPIRICAL AND
EXPECTED RESULTS
ARE THE SAME
THE LAW OF LARGE NUMBERS

T
he law of large numbers is According to the law, as you make
IN CONTEXT one of the foundations of more observations of an event
probability theory and occurring, the measured probability
KEY FIGURE
statistics. It guarantees that, over (or chance) of that outcome gets
Jacob Bernoulli (1655–1705)
the long term, the outcomes of ever closer to the theoretical
FIELD future events can be predicted chance as calculated before any
Probability with reasonable accuracy. This, for observations began. In other words,
example, gives financial companies the average result from a large
BEFORE the confidence to set prices for number of trials will be a close
c. 1564 Gerolamo Cardano insurance and pension products, match to the expected value as
writes Liber de ludo aleae (The knowing their chances of having to calculated using probability
Book on Games of Chance), pay out, and ensures that casinos theory – and increasing the number
the first work on probability. will always make a profit from their of trials will result in that average
gambling customers – eventually. becoming an even closer match.
1654 Pierre de Fermat and
Blaise Pascal develop
probability theory.
AFTER The expected chance of a random event can be
1733 Abraham de Moivre calculated using probability theory.
proposes what becomes the
central limit theorem – as
a sample size increases, the
results will more closely match As the In tests,
number of trials the observed results
normal distribution, or the
increases, the average do not closely match
bell curve. observed value gets closer the expected value
1763 Thomas Bayes develops to the expected one. straight away.
a way of predicting the chance
of an outcome by taking into
account the starting conditions
related to that outcome. After a large number of trials, the
average observed value and the expected
value are almost identical.
 THE ENLIGHTENMENT 185
See also: Probability 162–65 ■ Normal distribution 192–93 ■ Bayes’ theorem
198–99 ■ The Poisson distribution 220 ■ The birth of modern statistics 268–71

the probability of f being greater or

less than W by a specified amount
approached 0 (meaning impossible)
as the game was repeated.

We define the art The false probability

of conjecture… as the art of A coin toss is an example of the law
evaluating… the probabilities of large numbers. Assuming that
of things, so that in our the chance of a heads or tails result
judgments and actions we can is equal, the law dictates that after
always base ourselves on what many tosses, half (or very near it) Jacob Bernoulli
has been found to be the best. will have landed on heads, and
Jacob Bernoulli half on tails. However, in the early Born in Basel, Switzerland, in
stages, heads and tails are likely to 1655, Jacob Bernoulli studied
be more unbalanced. For example, theology, but developed an
the first 10 tosses could be seven interest in mathematics. In
heads and three tails. It might then 1687, he became a professor
seem most likely that the next toss of mathematics at the
will produce a tail. That, however, University of Basel, a position
The law was named by French is the “gambler’s fallacy” – where a he held for the rest of his life.
mathematician Siméon Poisson person assumes that the outcomes In addition to his work
in 1835, but its origin is credited of each game (toss) are connected. on probability, Bernoulli is
remembered for discovering
to Swiss mathematician Jacob A gambler might assume that toss
the mathematical constant e
Bernoulli. His breakthrough, which number 11 is likely to be a tail
by calculating the growth of
he called the “golden theorem”, was because the number of heads
funds that received compound
published by his nephew in 1713 in and tails must balance out, interest continuously in
the book Ars Conjectandi (The Art but the probability of heads or infinitesimal increments.
of Conjecturing). tails is the same in every toss, He was also involved in the
Although not the first person to and the outcome of one toss occurs development of calculus,
recognize the relationship between independently of any other. This is taking the side of Gottfried
collecting data and predicting the starting point of all probability Leibniz against Isaac Newton
results, Bernoulli developed the theory. After 1,000 tosses, the in their rival claims to have
first proof of this relationship by imbalance apparent in those first invented a new mathematical
considering a game with two 10 tosses becomes negligible. ■ field. Bernoulli worked on
possible outcomes – a win or a loss. calculus with his younger
The theoretical chance of winning brother Johann. However,
the game is W, and Bernoulli Johann became jealous of his
suspected that the fraction of brother’s achievements and
their relationship broke down
games (f ) that resulted in a win
several years before Jacob
would converge on W as the died in 1705.
number of games increased.
He proved this by showing that
Key works
When a referee flips a coin, there
is no advantage, according to the law 1713 Ars Conjectandi (The Art
of large numbers, in a team captain of Conjecturing)
basing a heads or tails choice on what 1744 Opera (Collected Works)
has been called in previous games.
ONE OF THOSE
STRANGE
NUMBERS THAT ARE
CREATURES
OF THEIR OWN
EULER’S NUMBER
188 EULER’S NUMBER

IN CONTEXT A mathematical constant is a significant, well-defined

number. Its magnitude never varies.
KEY FIGURE
Leonhard Euler (1707–83)
FIELD
Number theory
The constant e (2.718…) has special properties.
BEFORE
1618 Logarithms calculated
from the number now known
as e are listed in an appendix
to a book on logarithms by It is irrational –
It is transcendental –
John Napier. it cannot be expressed as
it is still irrational when
a ratio of two integers
1683 Jacob Bernoulli uses e in raised to any power.
in a simple fraction.
his work on compound interest.
1733 Abraham de Moivre

T
discovers “normal distribution”:
the way that values for most he mathematical constant can be used to mathematically
data cluster at a central point that became known as e, describe many processes in nature,
and taper off at the extremes. or Euler’s number – 2.718… but with algebraic notation still in
to an infinite number of decimal its infancy, Napier saw logarithms
Its equation involves e.
places – first appeared in the early only as an aid to calculation
AFTER 17th century, when logarithms were involving the ratio of distances
1815 Joseph Fourier’s proof invented to help simplify complex covered by moving points.
that e is irrational is published. calculations. Scottish mathematician In the late 17th century, Swiss
John Napier compiled tables of mathematician Jacob Bernoulli used
1873 French mathematician logarithms to base 2.718…, which 2.718… to calculate compound
Charles Hermite proves that worked particularly well for interest, but it was Leonhard Euler,
e is transcendental. calculations involving exponential a student of Bernoulli’s brother
growth. These were later dubbed Johann, who first called the number
“natural logarithms” because they e. Euler calculated e to 18 decimal

Leonhard Euler Born in 1707, in Basel, Switzerland, among other fields. This was
Euler grew up in nearby Riehen. despite steadily losing his sight
Taught initially by his father, a from 1738 and becoming blind in
Protestant minister who had some 1771. Working to the very end,
mathematical training and was he died in 1783 in St Petersburg.
also a friend of the Bernoulli
family, Euler developed a passion
for mathematics. Although he Key works
entered university to study for
the ministry, he switched to 1748 Introductio in analysin
mathematics with the support of infinitorium (Introduction to
Johann Bernoulli. Euler went on Analysis of the Infinite)
to work in Switzerland and Russia, 1862 Meditatio in experimenta
and became the most prolific explosione tormentorum nuper
mathematician of all time, instituta (Meditation upon
contributing greatly to calculus, experiments made recently
geometry, and trigonometry, on the firing of Cannon)
 THE ENLIGHTENMENT 189
See also: Positional numbers 22–27 ■ Irrational numbers 44–45 ■ Calculating pi 60–65 ■ Decimals 132–37
■ Logarithms 138–41 ■ Probability 162–65 ■ The law of large numbers 184–85 ■ Euler’s identity 197

places, writing his first work on e, interest rate of 3% per year would  0.25, so the investment after a
the Meditatio (Meditation), in 1727. produce £100  1.03  £103 after year would be £100  1.002512 
However, it was not published until one year. After two years, it would £103.04. If interest is calculated
1862. Euler explored e further in be 100  1.03  1.03  £106.09, daily, the rate is 3  365  0.008…
his 1748 Introductio (Introduction). and after 10 years it would be £100 and the amount after one year is
 1.0310  £134.39. The formula for £100  1.00008…365  £103.05. The
Compound interest
t r nt
this is A  P (1  r) , where A is formula for this is A  P (1  /n) ,
One of the earliest appearances the final amount, is the original
P where is the number of times the
n
of e was in calculating compound investment (principal), r is the interest is calculated in each year.
interest – where the interest on interest rate (as a decimal), and t As the time intervals at which
a savings account, for example, is the number of years. interest is calculated get smaller,
is paid into the account to increase If interest is calculated more the amount of interest yielded
the amount saved, rather than often than annually, the calculation at the end of a year approaches
being paid out to the investor. If the changes. For example, if interest is A  Per. Bernoulli came close to
interest is calculated on a yearly calculated monthly, the monthly working this out in his calculations,
basis, an investment of £100 at an rate is 1 ⁄12 of the yearly rate. 3  12 when he identified e as the limit ❯❯

Compounding interest yields a bigger total sum. The examples below show
how a £10 principal investment accrues interest if the yearly interest rate is
100 per cent, versus compound interest paid at shorter intervals.

1 year, 6 months, 3 months,

100% interest rate 50% interest rate 25% interest rate

January £10 principal deposit £10 principal deposit £10 principal deposit
February
March

April £10  0.25  £2.50

£10  £2.50  £12.50
May
June

July £10  0.5  £5 £12.50  0.25  £3.125

£10  £5  £15 £12.50  £3.125  £15.625
August
September

October £15.625  0.25  £3.906

£15.625  £3.906  £19.531
November
December

January £10  1  £10 £15  0.5  £7.50 £19.531  0.25  £4.883

£10  £10  £20 £15  £7.50  £22.50 £19.531  £4.883  £24.41
190 EULER’S NUMBER
The factorial of an integer is the
y product of the integer and all the
ex integers below it: 2 (2  1), 3 (3  2
 1), 4 (4  3  2  1), 5 (5  4  3
 2  1) and so on, adding one
Tangent
more term in the product each time. For the sake of brevity,
This can be shown as e  1 + 1 + we will always represent
(0,1) x 1 ⁄2! + 1 ⁄ 3! +1 ⁄4! in factorial notation.
this number, 2.718281828…
Euler calculated e to 18 decimal by the letter e.
places, but noted that the decimals Leonhard Euler
continued indefinitely. This means
that e is irrational. In 1873, French
The exponential function can be mathematician Charles Hermite
used to calculate compound interest. x
proved that e is also non-algebraic –
The function produces the curve y  e ,
which cuts the y axis at (0,1), and gets it is not a number with a terminating
exponentially steeper. This graph also decimal that can be used in a
shows the tangent to the curve. regular polynomial equation. rate, leading to ever-larger totals.
This makes it a “transcendental” Constant population growth can
number – a real number that cannot be calculated with the formula
n P  P0ert where P0 is the original
of (1 + 1/n) as n approaches infinity be computed by solving an equation.
n
(n  ∞). The formula (1 + 1/n) population number, r is the growth
gives closer values for e as n The growth curve rate, and t is time.
increases. For example, n  1 gives Compound interest is an example Plotted on a graph, e shows
a value for e of 2, n  10 gives a of exponential growth. Such growth other special properties. The graph
x
value for e of 2.5937… and n  100 can be plotted on a graph and will of y  e (the exponential function,
gives a value for e of 2.7048…. appear as a curve. In the 17th see left) is a curve whose tangent
When Euler calculated a value century, English cleric Thomas (the straight line that touches but
for e correct to 18 decimal places, Malthus posited that population does not intersect the curve) at the
he probably used the sequence e = also increases exponentially if there coordinates (0,1) also has a gradient
1 + 1 + 1 ⁄2 + 1 ⁄6 + 1 ⁄24 + 1 ⁄120 + are no checks on its growth, such (steepness) of precisely 1. This is
1 ⁄ 720, going up to 20 terms. He as war, famine, or food shortages. because the derivative (rate of
x x
arrived at these denominators by This means that the population change) of e is, in fact, e , and
using the factorial for each integer. continues to grow at the same the derivative is used to find the

The catenary mathematician Christiaan

Huygens – who coined the name
Sometimes defined as the shape catenary from the Latin catena
a hanging chain takes if it is only (“chain”) in 1690 – showed that,
supported at its ends, a catenary unlike a parabola, a catenary
is a curve with the formula y = 1 ⁄2 curve could not be given by
x x
 (e + e- ). Catenaries are often a polynomial equation. Three
found in nature and in technology. mathematicians – Huygens,
For example, a square sail under Gottfried Leibniz, and Johann
pressure from the wind takes the Bernoulli – each calculated a
form of a catenary. Arches in the formula for the catenary, coming
shape of an inverted catenary are to the same conclusion. Their
often used in architecture and results were published together
The Gateway Arch in St. Louis, construction due to their strength. in 1691. In 1744, Euler described
Missouri, USA is a flattened catenary For a long time, the catenary’s a catenoid – shaped like a waisted
arch, designed by Finnish-American shape was believed to be the cylinder and produced by rotating
architect Eero Saarinen in 1947. same as that of a parabola. Dutch a catenary about an axis.
 THE ENLIGHTENMENT 191
tangent. The tangent is used to
calculate the rate of change at a
specific point on a curve. Because
x
the derivative is e , the slope (a
measure of direction and steepness)
of the tangent line will always be
the same as the y value.

Derangements
The various ways in which a set
of items can be ordered are called
permutations. For example, the set
1, 2, 3 can be arranged as 1, 3, 2, or
2, 1, 3, or 2, 3, 1, or 3, 1, 2, or 3, 2, 1.
There are six total ways, including
the original, as the number of
permutations in a set is equal to
the factorial of the highest integer,
in this case 3! (short for 3  2  1).
Euler’s number is also significant
in a type of permutation called a
derangement. In a derangement,
none of the items can remain in
their original position. For four five items, the total number of To carbon-date organic material,
items, the number of possible permutations is 120, and with six it researchers test a sample – here from
permutations is 24, but to find the is 720, making the task of finding an ancient human bone – and use
derangements of 1, 2, 3, 4, all other all derangements a substantial one. Euler’s number to calculate its age
from the rate of radioactive decay.
arrangements beginning with 1 Euler’s number makes it
must first be eliminated. There are possible to calculate the number
three derangements starting with of derangements in any set. This With the number 10, partitions
2: 2, 1, 4, 3; 2, 3, 4, 1; and 2, 4, 1, 3. number equals the number of include 3 and 7, with a product
There are also three derangements permutations divided by e, rounded of 21; or 6 and 4 to produce 24;
starting with 3 and three starting to the nearest whole number. For or 5 and 5 to give 25, which is the
with 4, making nine in total. With example, for the set of 1, 2, 3, maximum product for a partition of
where there are six permutations, 10 using two numbers. With three
6  e  2.207… or 2, to the nearest numbers, 3, 3, 4 has a product of 36,
whole number. Euler analysed but moving into fractional numbers,
derangements of 10 numbers for 31 ⁄3  31 ⁄3  31 ⁄3  1000 ⁄27  37.037…
Frederick the Great of Prussia, who the largest for three numbers. For a
hoped to create a lottery to pay off four-way partition, 21 ⁄2  21 ⁄2  21 ⁄2
[Frederick the Great is] his debts. For 10 numbers, Euler  21 ⁄2  39.0625, but in a five-way
always at war; in summer found that the probability of getting split, 2  2  2  2  2  32. In
with the Austrians, in winter a derangement is 1 ⁄e to an accuracy short, (10 ⁄2)2  25, (10 ⁄3)3  37.037...,
with mathematicians. of six decimal places. (10 ⁄4)4  39.0625, and (10 ⁄5)5  32.
Jean le Rond d’Alembert This smaller result for a five-way.
French mathematician Other uses partition suggests that the optimal
Euler’s number is relevant in many number of splits for 10 is between 3
other calculations – for example, in and 4. Euler’s number can help to
splitting up (partitioning) a number find both the maximum product, as
to discover which numbers in the e(10/e)  39.598…, and number of
partition have the largest product. partitions: 10/e  3.678…. ■
192

RANDOM VARIATION
MAKES A PAT
NORMAL DISTRIBUTION
TERN

I
n the 18th century, French 1743. When a coin is flipped,
IN CONTEXT mathematician Abraham de there are two possible outcomes:
Moivre made an important “success” and “failure”. This type
KEY FIGURES
step forward in statistics; building of test, with two equally likely
Abraham de Moivre
on Jacob Bernoulli’s discovery outcomes, is called a Bernoulli trial.
(1667–1754), Carl Friedrich of binomial distribution, de Binomial probabilities arise when
Gauss (1777–1855) Moivre showed that events a fixed number, n, of such Bernoulli
FIELDS cluster around the mean ( b on trials, each with the same success
Statistics, probability graph below). This phenomenon probability, p, are carried out and
is known as normal distribution. the total number of successes is
BEFORE Binomial distribution (used to counted. The resulting distribution
1710 British physician describe outcomes based on one of is written as b(n, p). Binomial
John Arbuthnot publishes two possibilities) was first shown by distribution b(n, p) can take
a statistical proof of divine Bernoulli in Ars Conjectandi (The values from 0 to n, centred on
providence in relation to the Art of Conjecturing), published in a mean of np.
number of men and women
in a population. Finding the mean
Mean In 1721, Scottish baronet Alexander
AFTER b (average) Cuming gave de Moivre a problem
1920 Karl Pearson, a British value
concerning the expected winnings
FREQUENCY

statistician, expresses regret in a game of chance. De Moivre

about describing the Gaussian concluded that it came down to
curve as the “normal curve” finding the mean deviation (the
because it gives the impression average difference between the
that all other probability a c overall mean and each value in
distributions were “abnormal”. a set of figures) of binomial
VALUE MEASURED distribution. He wrote up his
1922 In the US, the New results in Miscellanea Analytica.
York Stock Exchange The bell curve is a visual illustration De Moivre had realized that
introduces the use of normal of normal distribution. The highest binomial outcomes cluster around
distribution to model the point of the curve (b) represents the
mean, which the values cluster around. their mean – on a graph, they
risks of investments. Values become less frequent the plot an uneven curve that gets
further they are from the mean, so closer to the shape of a bell (normal
are least frequent at points a and c. distribution) the more data is
 THE ENLIGHTENMENT 193
See also: Probability 162–65 ■ The law of large numbers 184–85 ■ The fundamental theorem of algebra 204–09
■ Laplace’s demon 218–19 ■ The Poisson distribution 220 ■ The birth of modern statistics 268–71

collected. In 1733, de Moivre was

satisfied that he had found a simple Events cluster around the mean.
way of approximating binomial
probabilities using normal
distribution, thus creating a bell
curve for binomial distribution on
a graph. He wrote up his findings Normal distribution applies
as a short paper, then included it Binomial distribution
to continuous data, which applies to discrete data,
in the 1738 edition of his Doctrine can take any value within a meaning that it has
of Chances. given range. It produces a distinct values.
bell curve.
Using normal distribution
From the mid-18th century, the
bell curve cropped up as a model
for all kinds of data. In 1809, Carl De Moivre says that for
Friedrich Gauss pioneered a large enough sample size, the
normal distribution as a useful normal bell curve can be used to
statistical tool in its own right. estimate binomial distribution.
French mathematician Pierre-Simon
Laplace used normal distribution
to model curves for random errors,
such as measurement errors, in variation. The board consisted of Pearson – popularized the use
one of the first applications of a a triangular array of pegs through of the term “normal” to describe
normal curve. which beads dropped from top to what was also known as a
In the 19th century, many bottom, where they collected in “Gaussian” curve.
statisticians studied variation a series of vertical tubes. Galton Today, normal distribution is
in experimental results. British measured how many beads were widely used to model statistical
statistician Francis Galton used in each tube and described the data, with applications ranging
a device called the quincunx (or resulting distribution as “normal”. from population studies to
Galton board) to study random His work – along with that of Karl investment analysis. ■

Abraham de Moivre Born in 1667, Abraham de Moivre elected as a fellow of the Royal
was raised as a Protestant in Society in 1697. As well as his
Catholic France, and lived there work on distribution, de Moivre
until 1685, when Louis XIV was best known for his work on
expelled the Huguenots. Briefly complex numbers. He died in
imprisoned for his religious beliefs, London in 1754.
de Moivre emigrated to England
upon his release. He became a Key works
private mathematics tutor in
London. He had hoped for a 1711 De Mensura Sortis (On the
university teaching position but Measurement of Chance)
he still faced some discrimination 1721–30 Miscellanea Analytica
as a Frenchman in England. (Miscellany of Analysis)
Nevertheless, de Moivre 1738 The Doctrine of Chances
impressed and befriended many (1st edition)
eminent scientists of the time, 1756 The Doctrine of Chances
including Isaac Newton, and was (3rd edition)
194

THE SEVEN
BRIDGES OF
K NIGSBERG
GRAPH THEORY

IN CONTEXT
KEY FIGURE Euler’s graph theory A graph consists of a
focuses on the discrete set of points (called
Leonhard Euler (1707–83)
connections between nodes or vertices) connected by
FIELDS different points. arcs (curves or edges).
Number theory, topology
BEFORE
1727 Euler develops the
constant e, which is used in If a path reaches all nodes, travelling each arc only once,
describing exponential growth it is an Eulerian path.
and decay.
AFTER
1858 August Möbius extends
Euler’s graph theory formula An Eulerian path is
to surfaces that are joined to impossible in the case of the
form a single surface. Königsberg bridges.
1895 Henri Poincaré publishes
his paper Analysis situs, in

G
which graph theory is
raph theory and topology developed a new type of geometry
generalized to create a new
began with Leonhard to show that it was impossible to
area of mathematics known
Euler’s attempt to find a devise such a route. Distances
as topology (the study of
solution to a mathematical puzzle – between points were not relevant:
properties of geometrical whether it was possible to make the only thing that counted was the
figures that are not affected a circuit of the seven bridges in connections between points.
by continuous deformation). Königsberg (now Kaliningrad, Euler modelled the Königsberg
Russia) without crossing any bridge bridges problem by making each
twice. The river flowed around an of the four land areas a point (node
island and then forked. Realizing or vertex) and making the bridges
that the problem related to the arcs (curves or edges) that joined
geometry of position, Euler the various points. This gave him
 THE ENLIGHTENMENT 195
See also: Coordinates 144–51 ■ Euler’s number 186–91 ■ The complex plane 214–15 ■ The Möbius strip 248–49
■ Topology 256–59 ■ The butterfly effect 294–99 ■ The four-colour theorem 312–13

plotted using x and y axes. More them – for example, to represent the
generally, a graph consists of a different lengths of roads on a map.
discrete set of nodes (or vertices) A weighted graph is also called
connected by arcs (or edges). a network. Networks are used to
The number of arcs meeting at a model relationships between
Read Euler, read Euler. node is called its degree. For the objects in many disciplines –
He is our master Königsberg graph, A has degree including computer science,
in everything. 5 and B, C, and D each have degree particle physics, economics,
Pierre-Simon Laplace 3. A path that travels each arc cryptography, sociology, biology,
once and only once is called an and climatology – usually with a
Eulerian path (or a semi-Eulerian view to optimizing a particular
path if the start and end are at property, such as the shortest
different nodes). distance between two points.
The Königsberg bridges problem One application of networks is
can be expressed as the question: to address the so-called “travelling
a “graph” that represented the “Is there an Eulerian or a semi- salesperson problem”. This involves
relationships between the land Eulerian path for the graph of finding the shortest route for a
and the bridges. Königsberg?” Euler’s answer is salesperson to travel from their
that such a graph must have at home to a series of cities and back
First graph theorem most two nodes of odd degree, again. The puzzle was allegedly
Euler began from the premise that but the Königsberg graph has four first set as a challenge on the back
each bridge could be crossed only odd degree nodes. of a cereal box. In spite of advances
once and each time a land area was in computing, no method exists
entered it also needed to be exited, Network theory that guarantees always to find the
which required two bridges in order Arcs on a graph may be “weighted” best solution, because the time this
to avoid crossing any bridge twice. (given degrees of significance) by takes grows exponentially as the
Each land area therefore needed to assigning numerical values to given number of cities increases. ■
connect to an even number of
bridges, with the possible exception
The city of Königsberg had seven
of the start and finish (if they were bridges linking two parts of the city
different locations). However, in the to its two islands. Euler’s graph
graph representing Königsberg (see shows that it is impossible C Journey
right), A is the endpoint of five to construct a route that
bridges and B, C, and D are each visits each island and
the endpoint of three. A successful crosses each bridge
route needs land areas (nodes or only once.
vertices) to have an even number of
bridges (arcs) to enter and exit by. River
Only the start and end points can A
have an odd number. If more than D
two nodes have an odd number of
arcs, then a route using each bridge
only once is impossible. By
showing this, Euler provided the
first theorem in graph theory. Island
The word “graph” is most
often used to describe a Cartesian Bridge Island
system of coordinates with points B
196

EVERY EVEN INTEGER

IS THE SUM OF
TWO PRIMES
THE GOLDBACH CONJECTURE

I
n 1742, Russian mathematician conjecture is valid and no exception
IN CONTEXT Christian Goldbach wrote to will be found. Mathematicians,
Leonhard Euler, the leading however, require a definitive proof.
KEY FIGURE
mathematician of the time. Over centuries, different “weak”
Christian Goldbach
Goldbach believed he had observed versions of the conjecture have
(1690–1764)
something remarkable – that every been proved, but no one to date has
FIELD even integer can be split into two proved the strong conjecture, which
Number theory prime numbers, such as 6 (3 + 3) seems destined to defeat even the
or 8 (3 + 5). Euler was convinced brightest minds. ■
BEFORE that Goldbach was right, but he
c. 200 ce Diophantus of could not prove it. Goldbach also
Alexandria writes his proposed that every odd integer
Arithmetica in which he lays above 5 is the sum of three primes,
out key issues about numbers. and concluded that every integer
from 2 upwards can be created by
1202 Fibonacci identifies
adding together primes; these
what becomes known as
additional proposals are dubbed
the Fibonacci sequence “weak” versions of the original
of numbers. “strong” conjecture, as they would
1643 Pierre de Fermat follow naturally if the strong
pioneers number theory. conjecture were true.
Manual and electronic methods
AFTER have, as yet, failed to find any even
1742 Leonhard Euler refines number that does not conform to
the Goldbach conjecture. the original strong conjecture. In
2013, a computer tested every even
1937 Soviet mathematician number up to 4  1018 without
UCLA’s Terence Tao, winner of
Ivan Vinogradov proves the the Fields Medal in 2006 and the
finding one. The bigger the number, Breakthrough Prize in mathematics in
ternary Goldbach problem, the more pairs of primes can create 2015, published a rigorous proof of a
a version of the conjecture. it, so it seems highly likely that the weak Goldbach conjecture in 2012.

See also: Mersenne primes 124 ■ The law of large numbers 184–85
■ The Riemann hypothesis 250–51 ■ The prime number theorem 260–61
 THE ENLIGHTENMENT 197

THE MOST
BEAUTIFUL
EQUATION
F
ormulated by Leonhard
IN CONTEXT Euler in 1747, the equation
known as Euler’s identity,
KEY FIGURE ei + 1 = 0, encompasses the
Leonhard Euler (1707–83)
five most important numbers in
FIELD mathematics: 0 (zero), which is It is simple… yet incredibly
Number theory neutral for addition and subtraction; profound; it comprises the
1, which is neutral for multiplication five most important
BEFORE and division; e (2.718..., the number mathematical constants.
1714 Roger Cotes, the English at the heart of exponential growth David Percy
mathematician who proofread and decay); i (√-1, the fundamental British mathematician
Newton’s Principia, creates an imaginary number); and  (3.142...,
early formula similar to Euler’s, the ratio of a circle’s circumference
but using imaginary numbers to its diameter, which occurs in
and a complex logarithm (a many equations in mathematics
type of logarithm used when and physics). Two of these numbers,
the base is a complex number). e and i, were introduced by Euler
himself. His genius lay in combining real number with an imaginary
AFTER all five milestone numbers with one, such as a + bi, where a and b
1749 Abraham de Moivre uses three simple operations: raising a are any real numbers. When Euler
Euler’s formula to prove his number to a power (for example, 54, raised the constant e to the power of
theorem, which links complex or 5  5  5  5), multiplication, the imaginary number i multiplied
numbers and trigonometry. and addition. by , he discovered that it equals
–1. Adding 1 to both sides of the
1934 Soviet mathematician
Complex powers equation produces Euler’s identity,
Alexander Gelfond shows that Mathematicians such as Euler ei + 1 = 0. The equation’s
e is transcendental, that is, asked themselves if it would be simplicity has led mathematicians to
irrational and still irrational meaningful to raise a number to a describe it as “elegant”, a description
when raised to any power. complex power – a complex number reserved for proofs that are profound
being a number that combines a yet also unusually succinct. ■

See also: Calculating pi 60–65 ■ Trigonometry 70–75 ■ Imaginary and complex

numbers 128–31 ■ Logarithms 138–41 ■ Euler’s number 186–91
198

NO THEORY
IS PERFECT
BAYES’ THEOREM

IN CONTEXT Bayes’ theorem is used to calculate probabilities of events

KEY FIGURE based on prior knowledge.
Thomas Bayes (1702–61)
FIELD
Probability
Conditions related to the event can help us assess its
BEFORE
probability more accurately.
1713 Jacob Bernoulli’s
Ars Conjectandi (The Art of
Conjecturing), published after
his death, sets out his new
mathematical theory The theorem can be used to predict more
of probability. accurately the likelihood of future events.
1718 Abraham de Moivre
defines the statistical

I
independence of events in his n 1763, Richard Price, a Welsh variables increases, so their
book The Doctrine of Chances. minister and mathematician, observed average gets closer to
AFTER published a paper called “An their theoretical average. For
1774 In his Memoir on the Essay Towards Solving a Problem example, if you toss a coin for
in the Doctrine of Chances”. Its long enough, the number of times
Probability of the Causes of
author, the Reverend Thomas Bayes, it comes up heads will get closer
Events, Pierre-Simon Laplace
had died two years earlier, leaving and closer to half the total of
introduces the principle of
the paper to Price in his will. It was tosses – a probability of 0.5.
inverse probability. a breakthrough in the modelling of In 1718, Abraham de Moivre
1992 The International Society probability and is still used today grappled with the mathematics
for Bayesian Analysis (ISBA) in areas as diverse as locating lost underpinning probability. He
is founded to promote the aircraft and testing for disease. demonstrated that, provided the
application and development Jacob Bernoulli’s book Ars sample size was large enough, the
of Bayes’ theorem. Conjectandi (1713) showed that distribution of a continuous random
as the number of identically variable – people’s heights, for
distributed, randomly generated example – averaged out into a bell-
 THE ENLIGHTENMENT 199
See also: Probability 162–65 ■ The law of large numbers 184–85 ■ Normal distribution 192–93 ■ Laplace’s demon 218–19
■ The Poisson distribution 220 ■ The birth of modern statistics 268–71 ■ The Turing machine 284–89 ■ Cryptography 314–17

95 people do not have P( A)  P(BA) = P(AB)

the disease but test P(B)
positive (10% of 950
1,000 patients people without
are tested, of the disease). 0.05  0.9 = 0.045
whom 50 have 0.05  0.90.95  0.1 0.045  0.095
the disease. 45 people have
the disease and The likelihood of The likelihood Simplified fraction
test positive (90% being in the 5% of being in the 95%
of 50 people with with the disease without the disease
the disease). = 32.14%

5 people have the The likelihood of The likelihood of The likelihood

disease but do not being in the 90% with being in the 10% that you have
test positive (10% correct test results with inaccurate the disease if
of 50 people with test results you test positive
the disease).

If a disease affects 5 per cent of the population you test positive – P ( AB) – is 90 per cent.
(event A) and is diagnosed using a test with 90 However, Bayes’ theorem factors in the false
per cent accuracy (event B), you might assume results produced by the test’s 10 per cent
that the probability ( P) of having the disease if inaccuracy – P ( B).

shaped curve, later named the To show how to calculate inverse denote this, Bayes introduced
“normal distribution” by German probabilities, Bayes considered two “conditional probabilities”. These
mathematician Carl Gauss. interdependent events – “event A” are given as P ( AB), the probability
and “event B”. Each has a probability of A given B, and P ( BA), the
Working out probabilities of occurring – P( A) and P(B) – probability of B given A. Bayes
Most real-world events, however, with P for each being a number managed to solve the problem of
are more complicated than the toss between 0 and 1. If event A occurs, how all four probabilities related
of a coin. For probability to be it alters the probability of event B to one another with the equation:
useful, mathematicians needed to happening, and vice versa. To P( AB) = P( A)  P(BA)/P(B). ■
determine how an event’s outcome
could be used to draw conclusions Thomas Bayes Defence of the Mathematicians
about the probabilities that led to Against the Objections of the
it. This reasoning based on the The son of a Nonconformist Author of the Analyst, in which
causes of observed events – rather minister, Thomas Bayes was he defended Isaac Newton’s
than using direct probabilities, born in 1702 and grew up in calculus foundations against the
such as the 50 per cent chance of a London. He studied logic and criticisms of the philosopher
heads coin toss – became known theology at the University of Bishop George Berkeley. Bayes
as inverse probability. Problems Edinburgh and followed his was made a fellow of the Royal
that deal with the probabilities father into the ministry, Society in 1742 and died in 1761.
of causes are called inverse spending much of his life
probability problems and might leading a Presbyterian chapel Key works
involve, for example, observing a in Tunbridge Wells, Kent.
Although little is known of 1736 An Introduction to the
bent coin landing on heads 13
Bayes’ life as a mathematician, Doctrine of Fluxions, and a
times out of 20 and then trying to in 1736 he anonymously Defence of the Mathematicians
determine whether the probability published An Introduction to Against the Objections of the
of that coin landing on heads lies the Doctrine of Fluxions, and a Author of the Analyst
somewhere between 0.4 and 0.6.
200

SIMPLY
A QUESTION
OF ALGEBRA
THE ALGEBRAIC RESOLUTION OF EQUATIONS

P
olynomial equations innovations, including new types
IN CONTEXT involving numbers and a of numbers – such as negative and
single unknown quantity complex numbers – as well as
KEY FIGURE
( x, and powers of x such as x2 and modern algebraic notation and
Joseph-Louis Lagrange x3) are a powerful tool for solving group theory.
(1736–1813)
real-world problems. An example of
FIELD a polynomial equation is x2 + x + Searching for solutions
Algebra 41 = 0. While such equations can The Babylonians and ancient
be solved approximately by repeated Greeks used geometrical methods
BEFORE numerical calculations, solving to solve problems that are now
628 Brahmagupta publishes them exactly (algebraically) was usually expressed by quadratic
a formula for solving many not achieved until the 18th century. equations. In medieval times, more
quadratic equations. The quest led to many mathematical abstract algorithmic approaches
1545 Gerolamo Cardano
creates formulae for resolving
cubic and quartic equations.
Equations can be solved numerically but only some
1749 Leonhard Euler proves can be solved algebraically.
that polynomial equations
of degree n have exactly n
complex roots (where n = 2, 3,
4, 5, or 6).
If you can use a …and if these
AFTER finite number of operations only involve
1799 Carl Gauss publishes the operations, such as +, , whole numbers or
first proof of the fundamental ,  and √… fractions…
theorem of algebra.
1824 In Norway, Niels Henrik
Abel completes Paolo Ruffini’s
1799 proof that there is no …the equation can be
general formula for the solved algebraically.
quintic equation.
 THE ENLIGHTENMENT 201
See also: Quadratic equations 28–31 ■ Algebra 92–99 ■ The binomial theorem 100–01 ■ Cubic equations 102–05
■ Huygens’s tautochrone curve 167 ■ The fundamental theorem of algebra 204–09 ■ Group theory 230–33

were established, and by the These are the The fundamental theorem of algebra says that a cubic equation
coefficients of has three solutions – which are three numbers which, when
16th century, mathematicians the equation. each is used to replace , make the equation equal to zero.
x
knew certain relations between
the coefficients of a polynomial
equation and its roots, and had
devised formulae to solve cubic
(highest power 3) and quartic
equations (highest power 4). In the
m x3  n x2  px  q  0
17th century, a general theory of
polynomial equations, now called The highest power x is the An algebraic equation is made
the fundamental theorem of in this equation is variable of up of variables and coefficients.
x 3, so it is a cubic the equation.
algebra, took shape. This stated The highest power of the equation
equation. determines how many solutions it has:
that an equation of degree n (where in this case, there are three solutions.
n
the highest power of x is x ) has
exactly n roots or solutions, which
may be real or complex numbers. a related, lower-degree polynomial Within 20 years of Lagrange’s work,
equation whose coefficients were Italian mathematician Paolo Ruffini
Roots and permutations related to the coefficients of the began to prove that there was no
In his Reflections on the algebraic original equation with a striking general formula for the quintic
resolution of equations (1771), innovation – he considered the equation. Lagrange’s investigation
French-Italian mathematician possible permutations (reorderings) into permutations (and symmetries)
Joseph-Louis Lagrange introduced of the roots. Lagrange’s insight into formed the basis of the even more
a general approach for solving the symmetries that arose from abstract and general group theory
polynomial equations. His work these permutations showed why the advanced by French mathematician
was theoretical – he investigated cubic and quartic equations could Évariste Galois, who used it to
the structure of polynomial be solved by formulae, and showed prove why it is impossible to
equations to find the circumstances (due to the different permutations resolve equations of degree 5 or
under which a formula could be of symmetries and roots) why a higher algebraically – that is, why
found for solving them. Lagrange formula for the quintic equation there is no general formula for
combined the technique of using needed a different approach. solving such equations. ■

Joseph-Louis Lagrange Born Giuseppe Lodovico Lagrangia Euler’s recommendation, he was

in Turin in 1736, Lagrange made Director of Mathematics at
embraced his family’s French the Berlin Academy, and in 1787
heritage and went by the French he moved to the Académie des
version of his name. As a young Sciences in Paris. Despite being
mathematician, self-taught, he an academic and a foreigner,
worked on the tautochrone Lagrange survived the French
problem and developed a new Revolution and Reign of Terror,
formal method to find the function and died in Paris in 1813.
that solved such problems. At the
age of 19, he wrote to Leonhard Key works
Euler, who recognized his talent.
Lagrange applied his method, 1771 Reflections on the algebraic
which Euler named the calculus of resolution of equations
variations, to study a wide range 1788 Analytical Mechanics
of physical phenomena, including 1797 Theory of analytic
the vibration of strings. In 1766, at functions
202

LET US
GATHER FACTS
BUFFON’S NEEDLE EXPERIMENT

I
n 1733, the mathematician and The relationship with  can be used
IN CONTEXT naturalist George Leclerc, the in a number of probability problems.
Comte de Buffon, raised – and One example involves a quarter
KEY FIGURE
answered – a fascinating question. circle, inscribed in a square, which
Georges Louis Leclerc,
If a needle is dropped onto a series curves from the top left corner of
Comte de Buffon (1707–88) of parallel lines, all the same width the square to the bottom right (see
FIELD apart, what is the likelihood that below). The bottom horizontal edge
Probability the needle will cross one of the of the square is the x axis and the
lines? Now known as Buffon’s left vertical edge is the y axis, with a
BEFORE needle experiment, it was one of value of 0 in the lower left corner and
1666 Liber de ludo aleae (On the earliest probability calculations. 1 in the corners at each end of the
Games of Chance) by Italian curve. When two numbers between
mathematician Gerolamo An elegant illustration 0 and 1 are chosen at random as the
Cardano is published. Buffon originally used the needle x and y coordinates, whether the
experiment to estimate  (pi) – the point will lie inside the quarter
1718 Abraham de Moivre ratio of a circle’s circumference to circle (success) or outside it (failure)
publishes The Doctrine of its diameter. He did this by can be deduced by examining
Chances, the first textbook dropping a needle of length l many a2 + b2, where a is the x coordinate
on probability. times onto a series of parallel lines
AFTER distance d apart, where d is greater
1
1942–46 The Manhattan than the needle’s length (d > l).
Project, a US-led body for Buffon then counted the number of
times the needle crossed the line
developing nuclear weapons,
as a proportion of total attempts (p)
makes extensive use of Monte
and came up with the formula that y
Carlo methods (computational  is approximately equal to twice
processes that model risk by the length of the needle l, divided
generating random variables). by the distance (d) multiplied by the
Late 1900s Quantum Monte proportion of needles crossing the 0 1
Carlo methods are used to line:  ≈ 2l  dp. The probability of x
explore particle interactions the needle crossing a line can be Using pi, the probability of a randomly
in microscopic systems. calculated by multiplying each side chosen point falling within the quarter
of the formula by p, then dividing circle in this square can be calculated
each side by  to get p ≈ 2l  d. as roughly 78 per cent.
 THE ENLIGHTENMENT 203
See also: Calculating pi 60–65 ■ Probability 162–65 ■ The law of large
numbers 184–85 ■ Bayes’ theorem 198–99 ■ The birth of modern statistics 268–71

d
Georges-Louis Leclerc,
Comte de Buffon
Born in Montbard, France, in
1707, Georges-Louis Leclerc
was urged by his parents to
l pursue a career in law, but
was more interested in botany,
d = distance Buffon’s needle experiment demonstrated how medicine, and mathematics,
between the lines probability can be connected to pi. Buffon classed which he studied at the
l = length of the needle needles as “successful” (pink) if they crossed a University of Angers, France.
line when dropped, or “unsuccessful” (blue) if they At the age of 20, he explored
didn’t, then calculated the probability of “success”. the binomial theorem.
Independently wealthy,
Buffon was able to write and
and b is the y coordinate. The result Ulam and his colleagues for the
study tirelessly, corresponding
is > 1 for points outside the curve random sampling used during with many of the scientific
and < 1 for points within it. The secret work on nuclear weapons in elite of his day. His interests
point is chosen at random, so could World War II. Monte Carlo methods were wide-ranging, and his
be anywhere in the square. Points went on to be useful in modern output was immense – on
on the line of the quarter circle applications, especially once topics ranging from ship-
can be counted as a success. The computers made it far less time- building to natural history
chance of “success” is pr2 (the area consuming to repeat a probability and astronomy. The comte
of a circle)  4. If the radius is 1, experiment over and over again. ■ also translated a number
r2 = 1, so the area is just p; for a of scientific works.
quarter circle, divide p by 4 to get Appointed keeper of the
approximately 0.78. The whole area Jardin du Roi, the royal
is the area of the square, which is botanical gardens in Paris,
1  1 = 1, so the probability of in 1739, Buffon enriched its
collections and doubled its
landing in the shaded area is
size. He held the post until
approximately 0.78  1 = 0.78.
his death in Paris in 1788.
The Monte Carlo method
This problem is an example of a Key works
wider class of experiments that
In wind energy yield analysis, the 1749–1786 Histoire naturelle
employ a statistical approach predicted energy output of a wind farm (Natural History)
called the Monte Carlo method, during its lifetime is calculated, giving 1778 Les époques de la nature
a code name coined by Polish- different levels of uncertainty, by using (The Epochs of Nature)
American scientist Stanislaw Monte Carlo probability methods.
ALGEBRA
OFTEN GIVES
MORE THAN IS
ASKED OF HER
THE FUNDAMENTAL THEOREM
OF ALGEBRA
206 THE FUNDAMENTAL THEOREM OF ALGEBRA

A
n equation asserts that
IN CONTEXT one quantity is equal to
another, and provides a
KEY FIGURE
means of determining an unknown
Carl Gauss (1777–1855)
number. Since Babylonian times,
FIELD scholars have sought solutions to
Algebra equations, periodically encountering
seemingly insoluble examples. In the
BEFORE 5th century bce, Hippasus’s attempts
1629 Albert Girard states to resolve x2 = 2 and his realization
that a polynomial of degree n that √2 was irrational (neither a
will have n roots. whole number nor a fraction) are
said to have led to his death for
1746 The first attempt at a
betraying Pythagorean beliefs. Some
proof of the fundamental 800 years later, Diophantus had no Gerolamo Cardano encountered
theorem of algebra (FTA) is knowledge of negative numbers, so negative roots while working on cubic
made by Jean d’Alembert. could not accept an equation where equations in the 16th century. His
x is negative, such as 4 = 4x + 20, acceptance of these as valid solutions
AFTER was an important step in algebra.
1806 Robert Argand publishes where x is -4.
the first rigorous proof of the
FTA that allows polynomials Polynomials and roots involve powers of an unknown
with complex coefficients. In the 18th century, one of the most value, such as x2. The “root” of a
studied areas of mathematics polynomial equation is a specific
1920 Alexander Ostrowski involved polynomial equations. numerical value that will replace
proves the remaining These are often used to solve the unknown value to make the
assumptions in Gauss’s problems in mechanics, physics, polynomial equal 0. In 1629, French
proof of the FTA. astronomy, and engineering, and mathematician Albert Girard

1940 Hellmuth Kneser gives

the first constructive variant
of the Argand FTA proof that A polynomial equation is an expression made up
allows for the roots to be found. of variables (such as x and y) and coefficients (such as 4)
along with operations (such as + and -) to form an equation
(such as x2 + 4x - 12 = y).

A root is a number that replaces a

This method of solving variable (such as x = -6) to make
problems by honest the equation equal zero.
confession of one’s ignorance
is called Algebra.
Mary Everest Boole
British mathematician
All polynomial equations This is known as the
have roots that are either fundamental theorem
real or complex. of algebra (FTA).
 THE ENLIGHTENMENT 207
See also: Quadratic equations 28–31 ■ Negative numbers 76–79 ■ Algebra 92–99 ■ Cubic equations 102–05 ■ Imaginary
and complex numbers 128–31 ■ The algebraic resolution of equations 200–01 ■ The complex plane 214–15

Finding the root(s) of an equation

y y
Curve showing
Curve showing the equation
3 2
y = x + x - 3x
the equation
2 = 2-2
y x 2

1 The equation
The equation touches the x
x touches the x x axis in three
−2 −1 0 1 2 axis in two −2 0 2 places: it has
−1 places: it has three real roots.
two real roots.
−2 −2

An equation of power 2, such An equation of power 3, such

as y = x2 - 2, always has two real as y = x3 + x2 - 3x, always has
or complex roots. three real or complex roots.

showed that a polynomial of degree however, do not have real-number what is called a complex number.
n will have n roots. The quadratic roots. This was a problem faced by Once mathematicians had
equation x2 + 4x - 12 = 0, for Italian mathematician Gerolamo accepted the necessity of negative
example, has two roots, x = 2 and Cardano and his peers in the 16th and complex numbers for solving
x = -6, both of which produce the century; while working on cubic certain equations, the question
answer 0. It has two roots because equations, they found that some remained as to whether finding
of the x2 term – 2 is the equation’s of their solutions involved square roots of higher-degree polynomials
highest power. If any quadratic roots of negative numbers. This would require the introduction
equation is drawn on a graph (as seemed impossible, because a of yet new types of number. Euler
shown above), these roots can be negative number multiplied by and other mathematicians, ❯❯
easily found: they are where the itself produces a positive result.
line touches the x axis. Although The problem was solved in
his theorem was useful, Girard’s 1572 when another Italian, Rafael
work was hindered by the fact Bombelli, set out the rules for an
that he had no concept of complex extended number system that
numbers. These would be key to included numbers such as √-1
producing a fundamental theorem alongside the real numbers. In Imaginary numbers
of algebra (FTA) for solving all 1751, Leonhard Euler investigated are a fine and wonderful
possible polynomials. the imaginary roots of polynomials, refuge of the divine spirit.
and called √-1 the “imaginary Gottfried Leibniz
Complex numbers unit”, or i. All imaginary numbers
The collection of all positive and are multiples of i. Combining the
negative, rational and irrational real and the imaginary, such as
numbers together make up the a + bi (where a and b are any real
real numbers. Some polynomials, numbers, and i = √-1), creates
208 THE FUNDAMENTAL THEOREM OF ALGEBRA
notably Carl Gauss in Germany,
would seek to address this question,
eventually concluding that the roots
of any polynomial are either real
or complex numbers – no further
types of number are needed.

Early research
The FTA can be stated in a number
of ways, but its most common
formulation is that every polynomial
with complex coefficients will have
at least one complex root. It can
Carl Gauss also be stated that all polynomials
of degree n containing complex
Born in Brunswick, Germany, coefficients have n complex roots.
in 1777, Carl Gauss showed The first significant attempt at
his mathematical talents early: proving the FTA was made in 1746
aged only three, he corrected by French mathematician Jean le Jean d’Alembert was the first to
an error in his father’s payroll Rond d’Alembert in his “Recherches attempt to prove the FTA. In France,
calculations, and by the age of sur le calcul intégral” (“Research on it is called the d’Alembert–Gauss
five he was taking care of his theorem, acknowledging the influence
integral calculus”). D’Alembert’s of d’Alembert on Gauss.
father’s accounts. In 1795, he proof argued that if a polynomial
entered Göttingen University P(x) with real coefficients has a
and in 1798, he constructed a complex root, x = a + ib, then it satisfy his fellow mathematicians.
regular heptadecagon (a
also has a complex root, x = Later attempts to prove the FTA
polygon with 17 sides) using a - ib. To prove this theorem, he included those of Leonhard Euler
only a ruler and compasses –
the biggest advance in polygon used a complicated idea now and Joseph-Louis Lagrange. While
construction since Euclid’s known as “d’Alembert’s lemma”. useful to later mathematicians,
geometry some 2,000 years In mathematics, a lemma is an these were also unsatisfactory. In
earlier. Gauss’s Arithmetical intermediary proposition used to 1795, Pierre-Simon Laplace tried an
Investigations, written at the solve a bigger theorem. However, FTA proof using the polynomial’s
age of 21 and published in d’Alembert did not prove his lemma “discriminant”, a parameter
1801, was key to defining satisfactorily; his proof was correct, determined from its coefficients
number theory. Gauss also but contained too many holes to which indicates the nature of its
made advances in astronomy roots, such as real or imaginary.
(such as the rediscovery of the His proof contained an unproved
astroid Ceres), cartography, assumption that d’Alembert had
the study of electromagnetism, avoided – that a polynomial will
and the design of optical always have roots.
instruments. However, he kept
many of his ideas to himself; a There are only two kinds of
great number were discovered Gauss’s proof
certain knowledge: awareness In 1799, at the age of 21, Carl
in his unpublished papers
after his death in 1855.
of our own existence and the Friedrich Gauss published his
truths of mathematics. doctoral thesis. It began with
Jean d’Alembert a summary and criticism of
Key work
d’Alembert’s proof, among others.
1801 Disquisitiones Gauss pointed out that each of
Arithmeticae (Arithmetical these earlier proofs had assumed
Investigations) part of what they were trying to
prove. One such assumption was
 THE ENLIGHTENMENT 209
that polynomials of odd degree established that every real number
(such as cubics and quintics) equation would have a complex
always have a real root. This is true, number solution, he had not
but Gauss argued that the point considered equations built from
needed to be proved. His first proof complex numbers such as x2 = i.
was based on assumptions about In 1806, Swiss mathematician I have had my results
algebraic curves. Although these Jean-Robert Argand found a for a long time, but I do
were plausible, they were not particularly elegant solution. Any not yet know how to
rigorously proved in Gauss’s work. It complex number, z, can be written arrive at them.
was not until 1920, when Ukrainian in the form a + bi, where a is the Carl Gauss
mathematician Alexander Ostrowski real part of z and bi is the imaginary
published his proof, that Gauss’s part. Argand’s work allowed
assumptions could all be justified. complex numbers to be represented
Arguably, Gauss’s first, geometric geometrically. If the real numbers
proof suffered for being premature – are drawn along the x axis and the
in 1799, the concepts of continuity imaginary numbers are drawn
and of the complex plane, which along the y axis, then the whole with solving an equation built
would have helped him explain his plane between them becomes the from real numbers could be sure
ideas, had not yet been developed. realm of the complex numbers. that they would find their solution
Argand proved that the solution for within the realm of complex
Argand’s additions every equation built from complex numbers. These groundbreaking
Gauss published an improved proof numbers could be found among the ideas formed the foundations of
of the FTA in 1816 and a further complex numbers on his diagram complex analysis.
refinement at his golden jubilee and that there was therefore no Mathematicians since Argand
lecture (celebrating 50 years since need to extend the number system. have continued to work on proving
his doctorate) in 1849. Unlike his Argand’s was the first truly rigorous the FTA using new methods. In
first, geometric approach, his proof of the FTA. 1891, for example, German Karl
second and third proofs were more Weierstrass created a method –
algebraic and technical in nature. Legacy of the theorem now known as the Durand–Kerner
Gauss published four proofs of the The proofs by Gauss and Argand method, due to its rediscovery by
FTA, but did not fully resolve the established the validity of complex mathematicians in the 1960s – for
problem. He failed to address the numbers as roots of polynomials. simultaneously finding all of the
obvious next step: although he had The FTA stated that anyone faced roots of a polynomial. ■

Applications of the FTA After publishing their results,

they were told that their proof
Research on the fundamental settled a conjecture by South
theorem of algebra has led to Korean astrophysicist Sun Hong
breakthroughs in other fields. In Rhie. Her conjecture concerned
the 1990s, British mathematicians images of distant astronomical
Terrence Sheil-Small and Alan light sources. Massive objects in
Wilmshurst extended the FTA to the Universe bend passing light
harmonic polynomials. These may rays from distant sources in a
have an infinite number of roots, phenomenon called gravitational
but in some cases there are a lensing, creating multiple images
finite number. In 2006, American seen through a telescope. Rhie
mathematicians Dmitry Khavinson posited that there would be a
An Einstein ring, first discovered and Genevra Neumann proved maximum number of images
in 1998, is the deformation of light that there was an upper limit to produced; this turned out to be
from a source into a ring through the number of roots of a certain exactly the upper bound found
gravitational lensing. class of harmonic polynomials. by Khavinson and Neumann.
THE 19
CENTUR
1800–1900
 TH
URY
212 INTRODUCTION

Amateur mathematician The 2,000-year-old problem of The Poisson distribution

Jean-Robert Argand Charles Babbage lays Euclid’s parallel postulate is is established and
explores the idea of the groundwork for resolved by János Bolyai and is still used to model the
plotting complex calculators and Nicolai Lobachevsky, who prove the number of times an
numbers as eventually computers with validity of hyperbolic geometry event occurs within a
coordinates. his Difference Engine. for which it does not hold. fixed period.

1806 1822 1829–32 1837

1814 1829 1832

The notion of a deterministic Carl Gustav Jacob Jacobi Évariste Galois dies aged
Universe that can be predicted makes significant 20, having developed
with full knowledge of every particle strides for both maths group theory to aid his
is proposed by Pierre-Simon Laplace, and physics in his work work on polynomials.
and later named Laplace’s demon. on elliptic functions.

P
rogress in mathematics theoretical. This trend was fostered developed theories of hyperbolic
accelerated through the by the influential work of Carl geometry and curved spaces,
19th century, with science Friedrich Gauss, regarded by many resolving the problem of Euclid’s
and mathematics now becoming in the field as the greatest of all parallel postulate. This opened
respected academic studies. As the mathematicians. He dominated the up a completely new approach to
Industrial Revolution spread and study of mathematics for much of geometry, paving the way for the
1848’s “Year of Revolution” saw the first half of the century, making nascent field of topology (the study
nationalism surge across old contributions to the fields of of space and surfaces) which was
empires, there was a renewed drive algebra, geometry, and number also influenced by the possibility of
to understand the workings of the theory, and giving his name to more than three dimensions offered
Universe in scientific terms, rather such concepts as Gaussian by William Hamilton’s discovery
than through religion or philosophy. distribution, Gaussian function, of quaternions.
Pierre-Simon Laplace, for example, Gaussian curvature, and Gaussian Perhaps the best known
applied the theories of calculus to error curve. of the pioneers of topology is
celestial mechanics. He proposed August Möbius, inventor of the
a form of scientific determinism, New fields Möbius strip, which had the
saying that with the relevant Gauss was also a pioneer of non- unusual property of being a two-
knowledge of moving particles, Euclidean geometries, which dimensional surface with only
the behaviour of everything in the epitomized the revolutionary spirit one side. Non-Euclidean geometries
Universe could be predicted. of 19th-century mathematics. were further developed by Bernhard
Another characteristic of The subject was taken up by Riemann, who identified and
19th-century mathematics was its Nicolai Lobachevsky and János defined different types of geometry
increasing tendency towards the Bolyai, who independently in multiple dimensions.
 THE 19TH CENTURY 213

Eugène Catalan
proposes his conjecture The Riemann
about powers of natural The term “matrix” hypothesis, which
numbers – it is not proven is coined by James remains unproven today,
for more than 150 years. Joseph Sylvester. is proposed.

1844 1850 1859

1843 1847 1858 1874

William Hamilton develops his George Boole uses The mathematical properties Georg Cantor is
idea of quaternions, which algebra as the basis of the Möbius strip are the first mathematician
will be crucial to the for his groundbreaking investigated by August to provide mathematical
technological developments mathematical logic. Ferdinand Möbius and precision for infinity.
of the next century. Johann Listing.

Riemann did not confine his another area of mathematics demand for a means of accurate
studied to geometry, however. that was becoming increasingly and quick calculation with his
As well as his work on calculus, abstract during the 19th century. mechanical calculating device, the
he made important contributions The groundwork for the growing “Difference Engine”. In so doing,
to number theory, following in the field of abstract algebra was laid he laid the groundwork for the
footsteps of Gauss. The Riemann by Évariste Galois, who, although invention of computers. Babbage’s
hypothesis, derived from the he died young, also developed work in turn inspired Ada Lovelace
Riemann zeta function concerning group theory while determining to devise the forerunner of modern
complex numbers, is as yet a general algebraic method for computer algorithms.
unsolved. Other notable discoveries solving polynomial equations. Meanwhile, there were other
in number theory at this time developments in mathematics
include the creation of set theory New technologies that were to have far-reaching
and the description of an “infinity of Not all mathematics in this period implications for later technological
infinities” of Georg Cantor, Eugène was purely theoretical – and even progress. Using algebra as his
Catalan’s conjecture about the some of the abstract concepts soon starting point, George Boole
powers of natural numbers, and the found more practical applications. devised a form of logic based on
application of elliptic functions to Siméon Poisson, for example, used a binary system, and using the
number theory proposed by Carl his knowledge of pure mathematics operators and, or, and not. These
Gustav Jacob Jacobi. to develop ideas such as the became the foundation of modern
Jacobi was, like Riemann, an Poisson distribution, a key concept mathematical logic, but just as
all-rounder, often linking different in the field of probability theory. importantly paved the way for the
fields of mathematics in new ways. Charles Babbage, on the other language of computers almost a
His primary interest was in algebra, hand, responded to practical century later. ■
214

COMPLEX NUMBERS
ARE COORDINATES
ON A PLANE
THE COMPLEX PLANE

IN CONTEXT Some equations cannot be solved without

KEY FIGURE using complex numbers.
Jean-Robert Argand
(1768–1822)
FIELD
Number theory Complex numbers have two components: a real number
and an imaginary number.
BEFORE
1545 Italian scholar Gerolamo
uses negative square roots to
solve cubic equations in his
book Ars Magna.
Real numbers (1, 0, Imaginary numbers can be
1637 French philosopher 1, etc.) are traditionally plotted on a line perpendicular
and mathematician René expressed on a to the number line – the two
Descartes develops a way to horizontal number line. lines make an x and y axis.
plot algebraic expressions as
coordinates on a grid.
AFTER
1843 Irish mathematician This creates a plane of complex numbers, with
William Hamilton extends the real numbers plotted along the x axis, and
complex plane by adding two imaginary numbers along the y axis.
more imaginary units to create
quaternions – expressions that

A
are plotted in a 4-D space. fter centuries of suspicion, mathematician Jean-Robert Argand
1859 By merging two complex mathematicians finally was to plot complex numbers
planes, Bernhard Riemann embraced the concept of (made up of a real and imaginary
develops a 4-D surface to help negative numbers in the 1700s. component) as coordinates on a
him analyse complex functions. They did so by using imaginary plane created by two axes – x for
numbers in algebra. In 1806, the real numbers and y for imaginary
key contribution of Swiss-born numbers. This complex plane
 THE 19TH CENTURY 215
See also: Quadratic equations 28–31 ■ Cubic equations 102–05 ■ Imaginary and complex numbers 128–31
■ Coordinates 144–51 ■ The fundamental theorem of algebra 204–09

polynomial equal to zero. Many y

seemingly simple polynomials, 5i

IMAGINARY NUMBERS
however, such as x2 + 1, do not 4i 3 + 5i
equal zero if x is a real number.
Plotting x2 + 1 on a graph with an x 3i
There can be very little… and y axis creates a neat curve that 2i
science and technology never passes through the origin, or i 7 + 2i x
that is not dependent on (0,0) point. To make the FTA work
complex numbers. for x2 + 1, Gauss and others used 0 1 2 3 4 5 6 7
Keith Devlin real numbers combined with REAL NUMBERS
British mathematician imaginary numbers to create
complex numbers. All numbers are An Argand diagram uses the x and y
in essence complex. For example, axes to represent real numbers and
imaginary numbers, combining them
the real number 1 is the complex
to plot complex numbers. This diagram
number 1 + 0i, while the number shows two numbers: 3 + 5i and 7 + 2i.
i is 0 + i. The equation x2 + 1 can
equal zero when x is i or i.
provided the first geometrical plane. So 1  i = i, which does not
interpretation of the distinctive Argand’s discovery appear on the real number x axis at
properties of complex numbers. As Argand began to plot complex all, but on the imaginary y axis.
numbers, he discovered that the Continuing to multiply by i results
Algebraic roots imaginary number i does not get in more 90° rotations, which is why
Imaginary numbers had emerged bigger if raised to higher powers. every four multiplications arrive
in the 16th century when Italian Instead, it follows a four-step pattern back at the start point.
mathematicians such as Gerolamo that repeats infinitely: i1 = i; i2 = –1; Plots of complex numbers –
Cardano and Niccolò Fontana i3 = –i, i4 = 1; i5 = i, and so on. This or Argand diagrams – make
Tartaglia found that solving cubic can be visualized on the complex complicated polynomials easier to
equations required a square root of plane. Multiplying real numbers solve. The complex plane is now a
a negative number. The square of a by imaginary numbers produces powerful tool that works far beyond
real number cannot be negative – 90° rotations through the complex the interests of number theory. ■
any real number multiplied by itself
results in a positive – so they Jean-Robert Argand journal in 1813, and in the next
decided to treat 1 as a new unit year he used the complex plane
that operated separately from the Little is known of Jean-Robert to produce the first rigorous
real numbers. Leonhard Euler first Argand’s early life. He was born proof of the fundamental
used i to denote the imaginary unit in Geneva in 1768, but appears theorem of algebra. Argand
(1) in his attempts to prove the to have had no formal education published eight more articles
fundamental theorem of algebra in mathematics. In 1806, he before his death in Paris in 1822.
(FTA). This theorem states that all moved to Paris to manage a
polynomial equations of degree n bookshop, and self-published the Key work
have n roots. This means that if x2 is work containing the geometrical
the highest power in an algebraic interpretation of complex 1806 Essai sur une manière
numbers for which he is known. de représenter les quantités
expression made up of a single
(Norwegian cartographer Casper imaginaires dans les
variable (such as x) and real
Wessel is now known to have constructions géométriques
coefficients (numbers multiplying used similar constructions in (Essay on a method of
the variable), the expression has a 1799.) Argand’s essay was representing imaginary
degree of two, and also two roots; republished in a mathematics quantities geometrically)
roots are values of x that make the
216

NATURE IS THE MOST

FERTILE SOURCE
OF MATHEMATICAL
DISCOVERIES
FOURIER ANALYSIS

T
he sound created by lower-pitched ones. In the 17th
IN CONTEXT vibrating strings has been century, Galileo recognized that
a topic of research for more sounds are produced by vibrations:
KEY FIGURE
than 2,500 years. In about 550 bce, the higher the frequency of the
Joseph Fourier (1768–1830)
Pythagoras discovered that if you vibrations, the higher the pitch
FIELD take two taut strings of the same of the sound we perceive.
Applied mathematics material and the same tension, but
one is twice the length of the other, Heat and harmony
BEFORE the short string will vibrate with By the end of the 17th century,
1701 In France, Joseph twice the frequency of the longer physicists including Joseph
Sauveur suggests that string and the resulting notes will Sauveur were making great strides
vibrating strings oscillate be an octave apart. in studying the relationships
with many waves of different Two centuries later, Aristotle between the waves in stretched
lengths at the same time. suggested that sound travelled strings and the pitch and frequency
through the air in waves, although of sounds that they produced.
1753 Swiss mathematician he incorrectly thought that higher- In the course of their research,
Daniel Bernoulli shows that pitched sounds travelled faster than mathematicians showed that any
a vibrating string consists
of an infinite number of
harmonic oscillations. This is the wave of the This is the wave of the
note A, which has a higher note E, which has
frequency of 220 Hertz. a frequency of 330 Hertz.
AFTER
1965 In the US, James Cooley
Amplitude (a)

Amplitude (a)

and John Tukey develop the

Fast Fourier Transform (FFT),
an algorithm that is able to 0.0 Time(t) 0.0 Time(t)
speed up Fourier analysis. 0.01 0.01

2000s Fourier analysis is used

to create a number of speech The graph of the wave The graph of the wave
recognition programs for
y = a sin (2 (220)t) =
y a sin (2 (330) )
t
computers and smartphones. Sounds are made of a complex series of tones. Fourier analysis can
separate out pure tones, which can be represented as sine waves on
a graph, from each other. Tones have frequency, which determines
pitch, and amplitude, which determines volume.
 THE 19TH CENTURY 217
See also: Pythagoras 36–43 ■ Trigonometry 70–75 ■ Bessel functions 221 ■ Elliptic functions 226–27 ■ Topology 256–59
■ The Langlands Program 302–03

Joseph Fourier Jean-Baptiste Joseph Fourier was where he pursued his studies in
born in Auxerre, France, in 1768. mathematical physics, including
A tailor’s son, he went to military work on the Fourier series
school, where his keen interest in (a series of sine waves that
mathematics led him to become a characterize sounds). In 1822,
successful teacher of the subject. Fourier was made the secretary
Fourier’s career was disrupted of the French Academy of
by two arrests – one for criticizing Sciences, a post he held until his
the French Revolution, the other death in 1830. Fourier is one of
for supporting it – but in 1798, 72 scientists whose names are
he accompanied Napoleon’s inscribed on the Eiffel Tower.
forces into Egypt as a diplomat.
Napoleon later made him a baron, Key works
and then a count. After Napoleon’s
fall in 1815, Fourier moved to 1822 Théorie analytique de la
Paris to become director of the chaleur (The Analytical Theory
Statistical Bureau of the Seine, of Heat)

string will support a potentially an object, at any time after a source wave can be understood in terms
infinite series of vibrations, starting of heat had been applied to one of of the amplitudes of its constituent
from the fundamental (the string’s its edges. sine waves, a set of numbers that
lowest natural frequency) and Fourier’s studies of heat is sometimes referred to as the
including its harmonics (integer distribution showed that no matter harmonic spectrum.
multiples of the fundamental). how complex a waveform, it could Today, Fourier analysis plays
The pure tone of a single pitch is be broken down into its constituent a key role in many applications
produced by a smooth repetitive sine waves, a process that is now including digital file compression,
oscillation called a sine wave called Fourier analysis. Since analysing MRI scans, speech
(see graph). The sound quality heat in the form of radiation is a recognition software, musical
of a musical instrument results wave, Fourier’s discoveries about pitch correction software, and
principally from the number and heat distribution had applications determining the composition
relative intensities of the harmonics to the study of sound. A sound of planetary atmospheres. ■
present in the sound, or its
harmonic content. The result is a
variety of waves interfering with
each other.
Joseph Fourier was attempting
to solve the problem of how heat
diffused through a solid object.
He developed an approach that
would allow him to calculate the
temperature at any location within

Fourier analysis of the way materials

vibrate allows engineers to construct
buildings that resonate at different
frequencies from a typical earthquake
and thus avoid the kind of damage that
occurred in Mexico City in 2017.
218

THE IMP THAT

KNOWS THE POSITIONS
OF EVERY PARTICLE
IN THE UNIVERSE
LAPLACE’S DEMON

I
n 1814, Pierre-Simon Laplace, exploration of determinism, a
IN CONTEXT a French mathematician who philosophical concept that
combined mathematics and says that all future events are
KEY FIGURE
science with philosophy and determined by causes in the past.
Pierre-Simon Laplace
politics, presented a thought
(1749–1827)
experiment now known as Mechanical analysis
FIELD Laplace’s demon. Laplace never Laplace was inspired by classical
Mathematical philosophy used the word “demon” himself; it mechanics – a field of mathematics
was introduced in later retellings, describing the behaviour of moving
BEFORE evoking a supernatural being made bodies, based on Isaac Newton’s
1665 Calculus is developed by godlike by mathematics. laws of motion. In a Newtonian
Isaac Newton to analyse and Laplace imagined an intellect universe, atoms (and even light
describe the motion of falling that could analyse movements of particles) follow the laws of motion,
bodies and other complex all atoms in the Universe in order and bounce around in a jumble of
mechanical systems. to accurately predict their future trajectories. Laplace’s “intellect”
paths. His experiment was an would be capable of capturing and
AFTER analysing all of their movements; it
1872 Ludwig Boltzmann uses would create a single formula that
statistical mechanics to show uses present movements to ascertain
how the thermodynamics of a past and predict future ones.
system always results in an Laplace’s theory had a startling
increase in entropy. philosophical consequence. It can
only work if the Universe follows a
1963 Edward Lorenz describes
predictable mechanical path, so
the Lorenz attractor, a model
that everything from the spin of
that produces chaotic results
galaxies to the tiny atoms in nerve
with every tiny change to the cells controlling thoughts could be
initial parameters. mapped out into the future. This
1872 American mathematician
David Wolpert disproves The orrery, a “clockwork universe”
Laplace’s demon by treating showing the movement of the celestial
the “intellect” as a computer. bodies in the Solar System, became a
popular device after the publication of
Newton’s universal theory of gravity.
 THE 19TH CENTURY 219
See also: Probability 162–65 ■ Calculus 168–75 ■ Newton’s laws of motion
182–83 ■ The butterfly effect 294–99

Is it possible to model the behaviour of every particle in

the Universe using classical mechanics (the motion of bodies
under the influence of forces)?

No, which means the

Yes, which means the Universe is probabilistic:
Universe is specific causes create
deterministic. Pierre-Simon Laplace
specific effects by chance.
Born into an aristocratic family
in 1749, Laplace lived through
the French Revolution and the
Reign of Terror, in which many
of his friends were killed. In
The future is already
The future is unwritten 1799, he became Minister of
determined and we have
and we have the ability to the Interior under Napoleon
no control over
influence it. Bonaparte, but was dismissed
our actions.
after only six weeks for being
too analytical and ineffectual.
Laplace later sided with the
Bourbons (the French royal
family) and was rewarded
with the return of his original
Laplace’s demon can title of marquis when the
Laplace’s demon
accurately predict monarchy was restored.
cannot exist.
the future. Laplace’s demon was a
side note to a career that also
encompassed physics and
astronomy, where Laplace
would mean that every aspect of a Instead, physicists used a technique was the first to postulate
person’s life up until their death has invented by Swiss mathematician the concept of a black hole.
already been predetermined; they Daniel Bernoulli in 1738, which His many contributions to
have no free will and no agency used probability theory to model mathematics were in classical
over their thoughts and deeds. the movement of independent units mechanics, probability theory,
and algebraic transformations.
within a space. Refined by Austrian
Laplace died in Paris in 1827.
Chance and statistics physicist Ludwig Boltzmann, this
Although mathematics helped to technique became known as
create such a crushing vision of statistical mechanics. It described
reality, it also helped to dismiss it. the atomic world in terms of Key works
By the 1850s, the study of heat and random chance – something at odds
1798–1828 Celestial
energy – thermodynamics – was with the mechanical determinism
Mechanics
ushering in a new model, the atomic of Laplace’s demon. By the 1920s,
1812 Analytic Theory
world. To do this, it needed to the idea of a probabilistic Universe of Probability
describe the motion of atoms and was solidified with the development 1814 A Philosophical Essay
molecules inside matter. Classical of quantum physics, which has on Probabilities
mechanics was not up to the task. uncertainty at its heart. ■
220

WHAT ARE THE

CHANCES?
THE POISSON DISTRIBUTION

I
n statistics, the Poisson is key. If l = 4 (the average number
IN CONTEXT distribution is used to model of potatoes ordered in one day), and
the number of times a the number of potato orders on any
KEY FIGURE
randomly occurring event happens one day is B, the probability that B
Siméon Poisson (1781–1840)
in a given interval of time or space. is less than or equal to 6 is 89 per
FIELD Introduced in 1837 by French cent, while the probability that B is
Probability mathematician Siméon Poisson, less than or equal to 7 is 95 per cent.
and based on the work of Abraham The chef must be at least 90 per
BEFORE de Moivre, it can help to forecast a cent sure that demand will be met,
1662 English merchant John wide range of possibilities. so n will be 7 here. ■
Graunt publishes Natural and Take, for example, a chef who
Political Observations upon the needs to forecast the number of
Bills of Mortality, marking the baked potatoes that will be ordered
birth of statistics. in her café. She needs to decide
how many potatoes to pre-cook
1711 Abraham de Moivre’s each day. She knows the daily
De Mensura Sortis (On the average order, and decides to
Measurement of Chance), prepare n potatoes where there is
describes what is later known at least 90 per cent certainty that
as the Poisson distribution. n will match demand.
AFTER To use the Poisson distribution
1898 Russian statistician to calculate n, conditions must be
met: orders must occur randomly,
Ladislaus Bortkiewicz uses the
singly, and uniformly – on average,
Poisson distribution to study
the same number of potatoes are
the number of Prussian ordered each day. If these conditions
soldiers killed by horse kicks. apply, the chef can find the value of
Siméon Poisson is credited with
finding the Poisson distribution, but
1946 British statistician R. D. n – how many potatoes to pre-bake. this may be an example of Stigler’s
Clarke publishes a study, based The average number of events per Law – no scientific discovery is
on the Poisson distribution, of unit of space or time (lambda, or l) credited to the true discoverer.
patterns of V-1 and V-2 flying
bomb impacts on London. See also: Probability 162–65 ■ Euler’s number 186–91 ■ Normal distribution
192–93 ■ The birth of modern statistics 268–71
 THE 19TH CENTURY 221

AN INDISPENSABLE
TOOL IN APPLIED
MATHEMATICS
BESSEL FUNCTIONS

I
n the early 19th century,
IN CONTEXT German mathematician and
astronomer Friedrich Wilhelm
KEY FIGURE
Bessel gave solutions to a particular
Friedrich Wilhelm Bessel
differential equation, the so-called
(1784–1846)
Bessel equation. He systematically Bessel’s functions are
FIELD investigated these functions very beautiful functions,
Applied geometry (solutions) in 1824. Now known as in spite of their having
Bessel functions, they are useful to practical applications.
BEFORE scientists and engineers. Central E. W. Hobson
1609 Johannes Kepler shows to the analysis of waves, such as British mathematician
that the orbits of the planets electromagnetic waves moving
are ellipses. along wires, they are also used to
describe the diffraction of light, the
1732 Daniel Bernoulli uses flow of electricity or heat in a solid
what later become known as cylinder, and the motions of fluids.
Bessel functions to study the
vibrations of a swinging chain. Movement of the planets breakthroughs in various fields.
1764 Leonhard Euler analyses The origins of Bessel functions lie Daniel Bernoulli found equations
a vibrating membrane using in the pioneering work of German for the oscillations of a pendulum,
what are later understood to mathematician and astronomer and Leonhard Euler developed
be Bessel functions. Johannes Kepler in the early 17th corresponding equations for the
century on the motions of the vibration of a stretched membrane.
AFTER planets. His meticulous analysis of Euler and others also used Bessel
1922 British mathematician observations led him to realize that functions to find solutions to the
George Watson writes his the orbits of the planets around the “three-body problem”, concerned
hugely influential A treatise on Sun are elliptical, not circular, and with the motion of a body, such
the theory of Bessel functions. he described the three key laws of as a planet or moon, being acted
planetary motion. Mathematicians upon by the gravitational fields
later used Bessel functions to make of two other bodies. ■

See also: The problem of maxima 142–43 ■ Calculus 168–75 ■ The law of large
numbers 184–85 ■ Euler’s number 186–91 ■ Fourier analysis 216–17
222
IN CONTEXT

IT WILL GUIDE
KEY FIGURES
Charles Babbage (1791–1871),
Ada Lovelace (1815–52)

THE FUTURE
FIELD
Computer science
BEFORE
1617 Scottish mathematician

COURSE OF
John Napier invents a manual
calculating device.
1642–44 In France,
Blaise Pascal creates a

SCIENCE
calculating machine.
1801 French weaver Joseph-
Marie Jacquard demonstrates
the first programmable
machine – a loom controlled

THE MECHANICAL COMPUTER

by a punchcard.
AFTER
1944 British codebreaker
Max Newman builds Colossus,
the first digital electronic
programmable computer.

B
ritish mathematician and
inventor Charles Babbage
anticipated the computer
age by more than a century
with two ideas for mechanical
calculators and “thinking”
machines. The first he called the
Difference Engine, a calculating
machine that would work
automatically, using a combination
of brass cogs and rods. Babbage
only managed to part-build the
machine, but even this was able
to process complex calculations
accurately in moments.
The second, more ambitious,
idea was the Analytical Engine. It
was never built, but was envisaged
as a machine that could respond
to new problems and solve them
without human intervention.
 THE 19TH CENTURY 223
See also: Binary numbers 176–77 ■ Matrices 238–41 ■ The infinite monkey theorem 278–79 ■ The Turing machine 284–89
■ Information theory 291 ■ The four-colour theorem 312–13

Charles Babbage was spurred

to start his work on a mechanical
calculator by the errors he found in
astronomical tables produced by
poorly paid and unreliable workers.
At each increase of
Engine was designed to handle knowledge, as well as on
numbers of up to 50 digits by the contrivance of every
means of more than 25,000 new tool, human labour
moving parts. becomes abridged.
To set up the machine for a Charles Babbage
calculation, each number was
represented by a column of
cogwheels, and each cogwheel
was marked with digits from 0 to 9.
A number was set by turning the
cogwheels in the column to show
The project received crucial input the correct digit on each. The calculating power. In 1823, he
from Ada Lovelace, a brilliant machine would then work through managed to persuade the British
young mathematician. Lovelace the entire calculation automatically. government to part-fund the
anticipated many of the key Babbage built several small project, with the promise that
mathematical aspects of computer working models with just seven it would make producing official
programming and foresaw how number columns but remarkable tables much quicker, cheaper, ❯❯
the machine could be used to
analyse any kinds of symbol.

Automatic calculation The Difference Engine used basic logic to perform…

In the 17th and 18th centuries,
mathematicians such as Gottfried
Leibniz and Blaise Pascal had
created mechanical calculating Iteration: the
aids, but these were limited Any arithmetical Any sequence repetition of any
in power and also prone to error as operations: of arithmetical sequence of
human input was needed at every +, , -,  operations. arithmetical
step. Babbage’s idea was to create operations.
a calculating machine that worked
automatically, eliminating human
error. He called his machine the
Difference Engine because it
allowed complex multiplications and If constructed, the Analytical Engine would be able to
divisions to be reduced to additions perform all of these, as well as…
and subtractions – “differences” –
that could be handled by scores
of interlocking cogs. It would even
print out the results.
No previous calculator had ever Conditional iterations: if P is an arithmetical operation and T is
worked with numbers larger than a test, then the engine iterates P until T fails.
four digits. Yet the Difference
224 THE MECHANICAL COMPUTER
This replica of the demonstration processes. The Mill was where
model Babbage made in 1832 of the arithmetical operations were
Difference Engine No. 1 has three performed; the Store was where
columns, each with its numbered
cogwheels. Two are for calculation,
numbers were held before
one for the result. processing and then received back
from the Mill after processing.
The Mill was Babbage’s version
Analytical Engine. His papers of a computer’s CPU, while the
suggest that the machine, if built, Store acted as its memory.
could have been close to what we The idea of telling a machine
now call a computer. His design what it should do – programming –
anticipated virtually all of the key came from a French weaver,
components of the modern Joseph-Marie Jacquard. He
computer, including the central developed a loom that used cards
processing unit (CPU), memory punched with holes to tell it how to
storage, and integrated programs. weave complex patterns in silk. In
One problem facing Babbage 1836, Babbage realized he too could
and more accurate. However, the was what to do with numbers use punched cards – to control his
full machine was hugely expensive carried over into the next column own machine but also to record
to develop, and tested the when adding up columns of digits. results and calculation sequences.
technological capability of the day At first, he used a separate
to its limits. After two decades’ mechanism for each carry-over, A supporting genius
work, the government cancelled but that proved too complicated. One of the greatest advocates for
the project in 1842. Then he split his machine into two Babbage’s work was his fellow
Meanwhile, in drawings and parts, the “Mill” and the “Store”, mathematician Ada Lovelace, who
calculations, Babbage had also which made it possible to separate wrote of the Analytical Engine that
been working on his idea for an the addition and carry-over it would “weave algebraic patterns

Types of punched card to program the Analytical Engine

Number cards Variable cards

specified the value of specified which data – held
numbers entered into the in “axes”, or storage units –
Store, or received numbers should be fed into the Mill,
back from the Store for and where returned data
external storage. should be stored.

Combinatorial cards
Operational cards
controlled how variable cards
determined the
and operational cards turned
arithmetical operations
backwards or forwards after
to be performed
specific operations
by the Mill.
were completed.
 THE 19TH CENTURY 225
as mechanical brains with wide
applications. “The engine can
arrange and combine its numerical
quantities exactly as if they were
letters or any other general
The object of the Analytical symbols”, she wrote, realizing that
Engine is twofold. First, any kind of symbol, not just
the complete manipulation numbers, could be manipulated
of number. Second, the and processed by machines.
complete manipulation of This is the difference between
algebraical symbols. calculation and computation – and
Charles Babbage the basis of the modern computer.
Lovelace also foresaw how such
machines would be limited by the Ada Lovelace
quality of the input. Arguably,
the first programmable computer – Born Augusta Byron in London
rather than calculator – was in 1815, Ada, Countess of
created by Konrad Zuse in 1938. Lovelace, was the only
just as the Jacquard loom weaves legitimate child of the poet
flowers and leaves”. As a teenager Delayed legacy Lord Byron. Byron left England
in 1832, Lovelace had seen one Lovelace’s plans to develop a few months after her birth,
of the Difference Engine models Babbage’s work were curtailed and Lovelace never saw her
working and had been instantly by her early death, by which time father again. Her mother, Lady
entranced. In 1843, she arranged Babbage himself was tired, ill, and Byron, was mathematically
gifted – Byron called her his
the publication of her translation disillusioned by the lack of support
“Princess of Parallelograms” –
of a pamphlet about the Analytical for his Difference Engine. The high-
and insisted Lovelace study
Engine written by Italian engineer precision mechanics required to mathematics too.
Luigi Menabrea, to which she build the machine were beyond Lovelace became renowned
added extensive explanatory notes. what any engineer could achieve for her talents in mathematics
Many of these notes covered at the time. Largely forgotten until and languages. She met
systems that would become part they were republished in 1953, Charles Babbage when she
of modern computing. In “Note G”, Lovelace’s notes confirm that she was 17 and was fascinated
Lovelace described possibly the and Babbage foresaw many of the by his work. Two years later,
first computer algorithm, “to show features of the computer now found she married William King,
an implicit function can be worked in every home and office. ■ Earl of Lovelace, with whom
out by the engine without human she had three children,
head and hands first”. She also but she continued to study
theorized that the engine could mathematics and follow the
solve problems by repeating a progress of Babbage, who
called her “the Enchantress
series of instructions – a process
of Number”.
known today as “looping”. Lovelace Lovelace wrote exhaustive
envisaged a program card, or set of The more I study [the
notes on Babbage’s Analytical
cards, that returned repeatedly to Analytical Engine], Engine. She set out many ideas
its original position to work on the the more insatiable I feel about what was to become
next data card or set. In this way, my genius for it to be. computing, earning herself a
Lovelace argued, the machine Ada Lovelace reputation as the first computer
could solve a system of linear programmer. Lovelace died in
equations or generate extensive 1852 of uterine cancer; in line
tables of prime numbers. Perhaps with her wishes, she was
the greatest insight in her notes buried next to her father.
was Lovelace’s vision of machines
226

A NEW KIND
OF FUNCTION
ELLIPTIC FUNCTIONS

IN CONTEXT Physics – to
KEY FIGURE calculate the charge
Carl Gustav Jacob Jacobi of a particle from its
(1804–51) curved path through a
magnetic field.
FIELDS
Astronomy – Mechanics – to
Number theory, geometry make calculations about
the orbits of planets
BEFORE are elliptical. the motion of
pendulums.
1655 John Wallis applies
calculus to the length of an
Some uses
elliptic curve; the elliptic
for elliptic
integral he derives is defined functions
by an infinite series of terms. include…
1799 Carl Gauss determines
the key characteristics of
elliptic functions, but his work Trigonometry – Cryptography – to
functions in spherical obscure the keys
is not published until 1841. trigonometry based on involved in encrypting
1827–28 Niels Abel the circle are special cases private information
independently derives and of elliptic functions. in public.
publishes the same findings
as Gauss.

T
AFTER he “squashed-circle” of creates an ellipse (and then open
1862 German mathematician an ellipse is one of the curves called a parabola and
Karl Weierstrass develops a most recognizable curves a hyperbola). An ellipse is a
general theory of elliptic in maths. Ellipses have a long closed curve that is defined as
functions, showing that they history in mathematics. They were the set of all points in a plane, the
can be applied to problems in studied by the ancient Greeks as sum of whose distances from two
both algebra and geometry. one of the conic sections. Slicing fixed points – each one called a
through a cone horizontally creates focus – is always the same number.
a circle; slicing at a steeper angle (A circle is a special ellipse with
 THE 19TH CENTURY 227
See also: Huygens’s tautochrone curve 167 ■ Calculus 168–75 ■ Newton’s laws
of motion 182–83 ■ Cryptography 314–17 ■ Proving Fermat’s last theorem 320–23

developed into the elliptic functions

and became a way of analysing
many kinds of complex curves and
oscillating systems beyond the
simple ellipse.
I learnt with as much
astonishment as satisfaction Practical applications
that two young geometers… In 1828, Norwegian Neils
succeeded in their own Abel and German Carl Jacobi,
individual work in again working independently,
considerably improving the showed wider applications for Carl Gustav Jacob
theory of elliptic functions. elliptic functions in both
Jacobi
Adrien-Marie Legendre mathematics and physics. For
example, these functions appear Born in Potsdam, Prussia,
in the 1995 proof of Fermat’s last in 1804, Carl Gustav Jacob
theorem, and the latest public-key Jacobi was initially tutored by
cryptography systems. Since an uncle. Having learned all
Abel died at 26, just months after that school could teach him by
making his major discoveries, the age of 12, he had to wait
just one central focus, not two.) many of these applications were until he was 16 to be allowed
In 1609, German astronomer and developed by Jacobi. Jacobi’s to attend Berlin University,
mathematician Johannes Kepler elliptic functions are complex, but and spent the intervening
demonstrated that the orbits of a more simple form, the p-function, years teaching himself
the planets were elliptical, with the was introduced in 1862 by German mathematics. He continued
to do so when he found the
Sun being located at one of the foci. mathematician Karl Weierstrass.
P -functions are used in classical university courses too basic.
He graduated within a year,
New tools and quantum mechanics. ■ and in 1832 he became a
Just as the mathematics of a circle professor at the University of
could be used to model and predict Königsberg. Falling ill in 1843,
natural phenomena that varied and Jacobi returned to Berlin,
repeated in a rhythmic (or periodic) where he was supported by
way, such as the up-and-down a pension from the king of
motion of a simple sound wave, Prussia. In 1848, he ran
the mathematics of the ellipse unsuccessfully for parliament
can be used to do the same for as a liberal candidate and the
phenomena that follow more offended king temporarily
complex periodic patterns, such as withdrew his support. In 1851,
electromagnetic fields or the orbital aged just 46, Jacobi contracted
smallpox and died.
motion of planets.
The genesis of such tools, the
elliptic functions, began in England
with 17th-century mathematicians
John Wallis and Isaac Newton. Key work
Working independently, they
Elliptic functions are used to define 1829 Fundamenta nova theoria
developed a method for calculating the trajectories of spacecraft such functionum ellipticarum (The
the arc length, or length of a as the Dawn probe, which explored the foundations of a new theory
section, of any ellipse. With later dwarf planet Ceres and the asteroid of elliptic functions)
contributions, their technique was Vesta in the asteroid belt.
228

I HAVE CREATED
ANOTHER WORLD
OUT OF NOTHING
NON-EUCLIDEAN GEOMETRIES

T
he parallel postulate (PP) there an obvious way of verifying
IN CONTEXT is the fifth of five postulates it. However, without the PP, many
from which Euclid deduced elementary theorems in geometry
KEY FIGURE
his theorems of geometry in his could not be proved. Over the next
János Bolyai (1802–60)
Elements. The PP was controversial 2,000 years, mathematicians would
FIELD among the ancient Greeks, since stake their reputations on attempts
Geometry it did not seem as self-evident as to resolve the issue. In the 5th
Euclid’s other postulates, nor was century ce, the philosopher Proclus
BEFORE
1733 In Italy, mathematician
Giovanni Saccheri fails to Euclidean and non-Euclidean geometries
prove Euclid’s parallel
postulate from his other In Euclidean geometry P
four postulates. (see right) the surface is B
assumed to be flat. In
1827 Carl Friedrich Gauss non-Euclidean forms of A
publishes his Disquisitiones geometry (see below), this is
generales circa superficies not the case. In hyperbolic The parallel postulate (PP) can be
curvas (General Investigations geometry, the surface curves expressed by Scottish mathematician John
inwards like a saddle, while Playfair’s Axiom: given a plane containing
of Curved Surfaces), defining an elliptic surface curves a line A and a point P not on A, there exists
the “intrinsic curvature” of a outwards like a sphere. exactly one line B through P that does not
space, which can be deduced intersect A. These lines A and B are parallel.
B
from within the space.
C
AFTER P
P
1854 Bernhard Riemann
B
describes the kind of surface A
that has hyperbolic geometry.
A
1915 Einstein describes In hyperbolic geometry, there are In elliptic geometry, such as on
gravity as curvature in infinitely many lines (e.g. B and C) the surface of a sphere, the PP does
spacetime in his general through point P that do not intersect not hold and every line (e.g. B) through
theory of relativity. line A. Surfaces in hyperbolic geometry point P intersects line A. For example,
exhibit “negative curvature” – for Earth’s meridians are parallel lines
example, the bell of a trumpet. that intersect at the poles.
 THE 19TH CENTURY 229
See also: Euclid’s Elements 52–57 ■ Projective geometry 154–55 ■ Topology
256–59 ■ 23 problems for the 20th century 266–67 ■ Minkowski space 274–75

time. Gauss acknowledged its

validity, but claimed to have
discovered it first. Gauss’s idea
of the “intrinsic curvature” of a
surface or space was an important
Leave the science of tool in establishing this new world,
parallels alone. I was ready but he left little evidence of having
to… remove the flaw from developed non-Euclidean geometry
geometry [but] turned back himself. He did, however, consider
when I saw that no man can that the Universe might be non-
reach bottom of this night. Euclidean. Subsequent advances Daina Taimina
Wolfgang Bolyai by Bernhard Riemann, Eugenio
Father of János Bolyai Beltrami, Felix Klein, David Hilbert, Born in Latvia in 1954, Daina
and others mean that today, non- Taimina began her career
Euclidean geometries are no longer in the fields of computer
seen as exotic, and physicists have science and the history of
given serious consideration to mathematics. After teaching
whether our Universe is indeed for 20 years at the University
flat (Euclidean) or curved. of Latvia, she moved to
argued that the PP was a theorem Cornell University in the
that could be derived from the Artistic explorations United States in 1996, where
other postulates and should Hyperbolic geometry also features a chance encounter opened
up a new area of interest.
therefore be struck out. in art. Models devised by Henri
Taimina attended a geometry
During the Golden Age of Islam Poincaré inspired many graphic
workshop led by David
(8th–14th century), mathematicians works by M. C. Escher, while some Henderson in which he
attempted to prove the PP. Persian mathematicians, notably Daina demonstrated how to make
polymath Nasir al-Din al-Tusi Taimina, have used art and craft a paper model of a hyperbolic
showed that the PP is equivalent techniques to make these “new surface. Henderson himself
to stating that the sum of angles worlds” intuitively graspable. ■ had learned the technique
in any triangle is 180°, but the PP from pioneering American
nonetheless remained controversial. topologist William Thurston.
In the 17th century, new translations Taimina went on to make
of Elements reached Europe, and her own models of hyperbolic
Giovanni Saccheri showed that if surfaces using crochet to
the PP was untrue, then the sum assist in her teaching. The
of angles in a triangle was always models were a success,
either less than or greater than 180°. breaking the stereotype
of mathematics as a field
By the early 19th century,
unrelated to arts and crafts.
Hungarian János Bolyai and Taimina has since embarked
Russian Nicolai Lobachevsky on a second career as a
independently proved the validity mathematician–artist.
of a “hyperbolic” non-Euclidean
geometry (see opposite) in which
the PP did not hold but the other
Crochet models of hyperbolic surfaces Key work
four of Euclid’s postulates did. created by Daina Taimina are more
Bolyai claimed to have “created tactile than paper models. She claims 2004 Experiencing Geometry
another world out of nothing”, but that the crocheting process helps with David W. Henderson
the idea was not well received in its develop geometrical intuition.
230
IN CONTEXT

ALGEBRAIC
KEY FIGURE
Évariste Galois (1811–32)
FIELDS

STRUCTURES
Algebra, number theory
BEFORE
1799 Italian mathematician
Paolo Ruffini considers the sets

HAVE
of permutations of roots as an
abstract structure.
1815 Augustin-Louis Cauchy,
a French mathematician,

SYMMETRIES
develops his theory of
permutation groups.
AFTER
1846 Galois’ work is published
posthumously by fellow

GROUP THEORY Frenchman Joseph Liouville.

1854 British mathematician
Arthur Cayley extends the
work of Galois to a full theory
of abstract groups.
1872 German mathematician
Felix Klein defines geometry
in terms of group theory.

G
roup theory is a branch
of algebra that pervades
modern mathematics.
Its genesis was largely due to
French mathematician Évariste
Galois, who developed it in order
to understand why only some
polynomial equations could be
solved algebraically. In so doing, he
not only gave a definitive answer to
a historical quest that had begun in
ancient Babylon, but also laid the
foundations of abstract algebra.
Galois’ approach to this problem
was to relate it to a question in
another area of mathematics. This
can be a powerful strategy when the
other area is well understood. In this
case, however, Galois first had to
 THE 19TH CENTURY 231
See also: The algebraic resolution of equations 200–01 ■ Emmy Noether and
abstract algebra 280–81 ■ Finite simple groups 318–19

A group is a set of elements, such as

numbers or shapes…

…along with an operation (such as addition or rotation) that

acts upon them.

Évariste Galois
Born in 1811, Évariste Galois
lived a brief but fiery and
To be labelled a group, a set must brilliant life. He was already
satisfy four axioms. familiar as a teenager with the
works of Lagrange, Gauss and
Cauchy, but failed (twice) to
enter the prestigious École
Polytechnique – possibly due
to his mathematical and
It must have an It must have an
political impetuousness,
identity: an element inverse: every element
though no doubt affected
that leaves any other has a corresponding
by the suicide of his father.
element unchanged element that combines
In 1829, Galois enrolled at
when it acts on it. to give the identity.
the École Préparatoire, only
to be expelled in 1830 for his
politics. A staunch republican,
he was arrested in 1831 and
It must be imprisoned for eight months.
It must be closed: Shortly after his release in
associative: the order
performing the 1832, he became involved in
in which the operations
operation will not a duel – it is unclear whether
are performed on
introduce elements this was over a love affair or
the elements does
outside of the set. politics. Badly wounded, he
not matter.
died the next day, leaving
behind just a handful of
mathematical papers which
contain the foundations of
develop the theory of the “simpler” subject to some axioms. When
group theory, finite field
area (the theory of groups) in order these elements include shapes, theory, and what is now
to tackle the more difficult problem groups can be thought of as called Galois theory.
(solubility of equations). The link he encoding symmetry. Simple
made between the two areas is now symmetries – such as those of a
called Galois theory. regular polygon – are intuitively Key works
graspable. For example, an
1830 Sur la théorie des
Arithmetic of symmetries equilateral triangle with the nombres (On Number Theory)
A group is an abstract object – it vertices A, B, and C (see next 1831 Premier Mémoire (First
consists of a set of elements and page) can be rotated in three Memoir)
an operation that combines them, ways (through 120°, 240°, or 360°) ❯❯
232 GROUP THEORY
e  2
A C B

 

C B B A A C

A B C

 

B C C A A B
  2
The equilateral triangle has six symmetries. They are symmetry after another to e, the identity element (rotation
rotation () through 120°, 240°, and 360° and reflection through 0°), and how they are written – 2  (the last
() through a vertical line through A, B, or C. The equilateral triangle in the diagram) means “rotate
diagram above shows the results of applying one through 120 degrees twice and reflect”.

about its centre, and be reflected in of a set of elements. The symmetry The second axiom is the inverse
three different lines. Each of these group of the equilateral triangle is a axiom. It says that every element
six transformations fits the triangle member of a small group called D3. has a unique inverse element;
onto itself – it looks exactly the combining the two yields the
same, except that the vertices are Axioms of group theory identity element.
permuted (rearranged). A clockwise Group theory has four main axioms. The third axiom concerns
rotation of 120° sends vertex A to The first is the identity axiom; it associativity, which means that
where B was, B to C, and C to A, states that a unique element exists the result of operations on elements
while a reflection in the vertical line that does not change any element does not depend on the order in
through A swaps vertices B and C. in the group when combined with which they are applied. For example,
The three rotations and the it. With the ABC triangle, the if you combine any set of three
reflections give all possible identity is the rotation of 0°. elements with a multiplication
symmetries of the triangle ABC. operator, you can perform the
One way to see the symmetries operations in any order. So if the
of the triangle is to consider all of elements 1, 2, and 3 are members of
the possible permutations of the a group, then (1  2)  3 = 2  3 =
vertices. A rotation or reflection 6, and 1  (2  3) = 1  6 = 6, all
can send the vertex A to one of giving the same result.
three points (including itself). The fourth axiom is closure,
From each of these possibilities, meaning that a group should have
the vertex B has two available no elements outside the group as a
destinations. The destination of
the third vertex is now determined
The possible rotations of a Rubik’s
because the triangle is rigid, so Cube form a mathematical group with
there are 3  2  6 possibilities. 43,252,003,274,489,856,000 elements,
The symmetry groups of polygons but solving the cube from any position
can be thought of as permutations requires no more than 26 turns of 90°.
 THE 19TH CENTURY 233

Group theory in physics In 1915, German algebraist

Emmy Noether demonstrated
The Universe, as we understand how Lie groups related to
it through physics, is full of conservation laws (such as
symmetries, and group theory is the conservation of energy).
proving a powerful tool for both By the 1960s, physicists began
understanding and prediction. to use group theory to classify
Physicists use the Lie groups, subatomic particles. But the
named after the 19th-century mathematical groups they used
Norwegian mathematician Sophus included a combination of
Lie. Lie groups are continuous, symmetries that no known
not discrete – for example, they particles had. Scientists tried
model the infinite number of looking for a particle with that
The ATLAS detector at the CERN rotational symmetries, such as combination of symmetries, and
accelerator is designed to study those associated with a circle, found the Omega minus particle.
subatomic particles, including rather than the finite number More recently, the Higgs boson
those predicted by group theory. of transformations of a polygon. has filled another such gap.

result of performing the operations. means that its elements can be equations and simpler polynomials
One example of a group obeying all swapped around without changing are solvable, but those of higher
four axioms is the set of integers the result. Integers added in any degree polynomials are not.
{…, -3, -2, -1, 0, 1, 2, 3, …} with order will give the same result Modern algebra is an abstract
the operation of addition. The (6 + 7 = 13 and 7 + 6 = 13), so the study of groups, rings, fields, and
unique identity element is 0, and set of integers with the operation other algebraic structures.
the inverse of any integer n is -n of addition is an Abelian group. Group theory continues to
as n + -n  0  -n + n. The develop in its own right and has
addition of integers is associative, Galois groups and fields many applications. Group theory
and the set is also closed, because Groups are just one kind of abstract is used to study symmetries in
adding any of the integers together algebraic structure among many. chemistry and physics, for example,
gives another integer. Closely related structures include and can be used in public key
Groups can also have a further rings and fields, which are also cryptography, which secures much
attribute known as commutativity. defined in terms of a set with of today’s digital communication. ■
If a group is commutative, it is operations and axioms. A field
known as an Abelian group. This contains two operations; complex
numbers (with the operations of
addition and multiplication) are a
field. The field of complex numbers
is the territory in which solutions We need a super-
to polynomial equations are found. mathematics in which
Wherever groups disclosed Galois theory relates the the operations are as
themselves, or could solvability of a polynomial equation unknown as the quantities
be introduced, simplicity (whose roots are elements of a field) they operate on… such a
crystallized out of to a group – specifically, to the
super-mathematics is the
permutation group that encodes
comparative chaos. Theory of Groups.
possible rearrangements of its roots.
Eric Temple Bell Galois showed that this group, now
Arthur Eddington
Scottish mathematician British astrophysicist
called a Galois group, must have
one kind of structure if the equation
is algebraically solvable, and a
different kind of structure if it is
not. Galois groups of quartic
234

JUST LIKE A
POCKET MAP
QUATERNIONS

A
n extension of complex direction in which a smartphone
IN CONTEXT numbers, quaternions are is pointing. Quaternions were
used to model, control, and the brainchild of William Rowan
KEY FIGURE
describe motion in three dimensions, Hamilton, an Irish mathematician
William Rowan Hamilton
which is essential in, for example, who was interested in how to
(1805–65) creating the graphics of a video model movement mathematically in
FIELD game, planning a space probe’s three-dimensional space. In 1843,
Number systems trajectory, and calculating the in a flash of inspiration, he realized

BEFORE
1572 Italy’s Rafael Bombelli
creates complex numbers Complex numbers (sums of real and imaginary numbers) have
by combining real numbers, two dimensions and describe motion in two dimensions.
based on the unit 1, with
imaginary numbers, based
on the unit i.
1806 Jean-Robert Argand
To describe motion in three dimensions, we need an
creates a geometrical extended version of complex numbers.
interpretation of complex
numbers by plotting them
as coordinates to create
the complex plane.
AFTER A three-dimensional number is not
1888 Charles Hinton invents sufficient to describe motion in three dimensions.
the tesseract, which is an
extension of the cube into
four spatial dimensions.
A tesseract has four cubes,
six squares, and four edges A full description of motion in three-dimensional
meeting at every corner. space requires a four-dimensional number,
or quaternion.
 THE 19TH CENTURY 235
See also: Imaginary and complex numbers 128–31 ■ Coordinates 144–51
■ Newton’s laws of motion 182–83 ■ The complex plane 214–15

Once complex numbers were

understood, the next challenge for
mathematicians was to create a
number that worked the same way
in a three-dimensional space. The
logical answer was to add a third
number line, j, which ran at 90
degrees to both the real and
imaginary number lines, but no one
could figure out how such a number
added, multiplied, and so on. William Rowan
Since quaternions can model and Hamilton
control the motion of objects in three Four dimensions
dimensions, they are particularly useful Hamilton’s solution was to add a Born in Dublin in 1805,
in virtual reality games. fourth non-real unit, k. This created Hamilton became interested
a quaternion, with a basic structure in mathematics from the age
of a + bi + cj + dk, where a, b, c, of eight after meeting Zerah
that the “third dimension problem” and d are real numbers. The two Colburn, a touring American
could not be solved with a three- additional quaternion units, j and k, mathematical child prodigy.
dimensional number, but needed a share similar properties to i and are At the age of 22, while still
four-dimensional one (a quaternion). imaginary. A quaternion can studying at Trinity College,
define a vector, or a line in three- Dublin, he was appointed
Movements and rotations dimensional space, and can both professor of astronomy
Complex numbers are two- describe an angle and direction of at the university and Royal
Astronomer of Ireland.
dimensional: they are made up of rotation around that vector. Like the
Hamilton’s expertise
a real and an imaginary part, for complex plane, simple quaternion in Newtonian mechanics
example, 1 + 2i. As a result, the two mathematics, combined with basic enabled him to calculate
parts of any complex number can trigonometry, offers a way to the paths of heavenly bodies.
act as coordinates, and the number describe all kinds of movements He later updated Newtonian
can be plotted on a surface or plane. within three-dimensional space. ■ mechanics into a system that
The two-dimensional complex enabled further advances to
plane extends the one-dimensional be made in electromagnetism
number line by combining real and quantum mechanics. In
numbers with imaginary units. The 1856, he tried to capitalize
plotting of complex numbers then on his skills by launching the
enables the calculation of motion icosian game, in which players
and rotation in two dimensions. An undercurrent of thought search for a path connecting
Any linear motion from point A to B was going on in my mind the points of a dodecahedron
which gave at last a result… without returning to the same
can be expressed as the addition
An electric circuit seemed to point twice. Hamilton sold
of two complex numbers. Adding
the rights to the game for £25.
more numbers creates a sequence close; and a spark flashed forth, He died in 1865, following
of movements across the plane. To the herald of many long years. a severe attack of gout.
describe rotation, complex numbers William Rowan Hamilton
are multiplied together. Every
Key works
multiplication by i, the imaginary
unit, results in a 90° rotation, and 1853 Lectures on Quaternions
a rotation of any other angle is due 1866 Elements of Quaternions
to some factor or fraction of i.
236

POWERS OF
NATURAL NUMBERS
ARE ALMOST
NEVER CONSECUTIVE
CATALAN’S CONJECTURE

M
any problems in number than 1. The solution is x = 3, m = 2,
IN CONTEXT theory are easy to pose, y = 2, and n = 3, since 32 - 23 = 1.
but extremely difficult In other words, squares, cubes, and
KEY FIGURE
to prove. Fermat’s last theorem, for higher powers of natural numbers
Eugène Catalan (1814–94)
example, remained a conjecture are almost never consecutive. Five
FIELD (unproven claim) for 357 years. hundred years before, Gersonides
Number theory Like Fermat’s conjecture, Catalan’s had proved a special case of the
conjecture is a deceptively simple claim. He used only powers of
BEFORE claim about powers of positive 2 and 3, solving the equations
c. 1320 French philosopher integers that was proved long after
n m m n
3 − 2 = 1 and 2 − 3 = 1. In
and mathematician Levi ben its initial statement. 1738, Leonhard Euler similarly
Gershon (Gersonides) shows In 1844, Eugène Catalan claimed proved a case in which the only
that the only powers of 2 and that there is only one solution to the powers allowed were squares and
3 that differ by 1 are 8 = 23
m n
equation x - y = 1, where x, y, m, cubes. Euler did this by solving
and 9 = 32. and n are natural numbers (positive the equation x2 − y3 = 1. This was
integers) and m and n are greater closer to Catalan's conjecture, but
1738 Leonhard Euler proves
that 8 and 9 are the only
consecutive square or
cube numbers. Using natural numbers (positive integers), the smallest
difference between two powers is 1.
AFTER
1976 Dutch number theorist
Robert Tijdeman proves that,
if more consecutive powers
exist, there are only a finite m n
Catalan expressed this as the formula x - y = 1,
number of them. where and must be greater than 1.
m n
2002 Preda Mihăilescu proves
Catalan’s conjecture, 158 years
after it was formulated in 1844.
There is only one solution to this equation
using natural numbers: 32 - 23 = 1.
 THE 19TH CENTURY 237
See also: Pythagoras 36–43 ■ Diophantine equations 80–81 ■ The Goldbach
conjecture 196 ■ Taxicab numbers 276–77 ■ Proving Fermat’s last theorem 320–23

Every power of 1 is 1.
12 = 1
Difference of 0
13 = 1

Difference of 3
22 = 4
This is the only instance
Difference of 4 where a difference of 1 is
23 = 8 found between a square
and a cubed number.
Difference of 1
32 = 9
Eugène Catalan
Difference of 7
42 = 16 Born in Bruges, Belgium, in
1814, Eugène Catalan studied
If squared and cubed under French mathematician
numbers are lined up in
Difference of 9 Joseph Liouville at the École
order of their values, the
52 = 25 difference between each Polytechnique in Paris.
Catalan was a republican from
value becomes clear. The
difference between 23 an early age and a participant
and 32 is 1, and Catalan’s in the 1848 revolution. His
Difference of 2 conjecture states that this political beliefs led to his
33 = 27 is the only pair of squares,
cubes, or higher powers
expulsion from a number of
academic posts.
that differ by 1. Catalan was particularly
interested in geometry and
combinatorics (counting and
did not allow for the possibility that Robert Tijdeman found an upper arranging), and his name is
larger powers or exponents could bound (maximum size) for x, y, m, associated with the Catalan
result in consecutive numbers. and n. This proved that there is numbers. This sequence (1, 2,
only a finite number of powers that 5, 14, 42…) counts, among
Becoming a theorem can be consecutive. The truth of other things, the ways that
Catalan himself said that he could Catalan’s conjecture could now be polygons can be divided
not prove his conjecture completely. tested by checking each of these into triangles.
Other mathematicians tackled the powers. Unfortunately, Tijdeman’s Although he considered
problem, but it was only in 2002 upper bound is astronomically himself French, Catalan won
that Romanian mathematician large, making such computation recognition in Belgium, where
Preda Mihăilescu solved the practically unfeasible even for he lived from his appointment
outstanding issues and turned modern computers. as professor of analysis at the
University of Liège in 1865
conjecture into theorem. Mihăilescu’s proof of Catalan’s
until his death in 1894.
It might seem that Catalan’s conjecture does not involve any
conjecture must be false, since such computation. Mihăilescu built
simple calculations quickly yield on 20th-century advances (by Ke Key works
examples of powers that are almost Zhao, J. W. S. Cassels, and others)
1860 Traité élémentaire des
consecutive. For example, 33 - 52 = that had proved m and n must be
séries (Elementary Treatise
2, and 27 - 53 = 3. On the other odd primes for any further solutions
m n on Series)
hand, even these near-solutions of x - y = 1. His proof is not 1890 Intégrales eulériennes
are rare. One approach to proving as formidable as Andrew Wiles’s ou elliptiques (Eulerian or
the conjecture appeared to involve proof of Fermat’s last theorem, Elliptic Integrals)
making many calculations: in 1976, but it is still highly technical. ■
238
IN CONTEXT

THE MATRIX
KEY FIGURE
James Joseph Sylvester
(1814–97)

IS EVERYWHERE
FIELDS
Algebra, number theory
BEFORE
MATRICES 200 bce The ancient Chinese
text The Nine Chapters on the
Mathematical Art presents a
method for solving equations
using matrices.
1545 Gerolamo Cardano
publishes techniques
using determinants.
1801 Carl Friedrich Gauss
uses a matrix of six
simultaneous equations to
compute the orbit of the
asteroid Pallas.
AFTER
1858 Arthur Cayley formally
defines matrix algebra, and
proves results for 2  2 and
3  3 matrices.

M
atrices are rectangular
arrays (grids) of elements
(numbers or algebraic
expressions), arranged in rows
and columns enclosed by square
brackets. The rows and columns
can be extended indefinitely, which
enables matrices to store vast
amounts of data in an elegant
and compact manner. Although
a matrix contains many elements,
it is treated like one unit. Matrices
have applications in mathematics,
physics, and computer science,
such as in computer graphics
and describing the flow of a fluid.
The earliest known evidence
for such arrays comes from the
ancient Mayan civilization of
 THE 19TH CENTURY 239
See also: Algebra 92–99 ■ Coordinates 144–51 ■ Probability 162–65 ■ Graph
theory 194–95 ■ Group theory 230–33 ■ Cryptography 314–17

The dimensions of a matrix are important, as operations such as

addition and subtraction require the matrices involved to have the
same dimensions. The 2  2 matrices below are square matrices,
meaning that they have the same number of rows as they have
columns. The graphic below shows how matrices are added
together by adding the elements in corresponding positions.

a b e f a+e b+f The result is also

+  a 2  2 matrix.
c d g h c+g d+h
James Joseph
Sylvester
Born in 1814, James Joseph
Central America, c. 2600 bce. Some elimination system introduced Sylvester began his studies at
historians believe the Maya people by German mathematician Carl University College London, but
manipulated numbers in rows and Gauss in the 19th century, which left when he was accused by
columns to solve equations, and is still used today for solving another student of wielding
cite grid-like decorations on their simultaneous equations. a knife. He then went to
monuments and priestly robes Cambridge and came second
as evidence. Others, however, Matrix arithmetic in the university examinations
doubt these patterns represent In 1850, British mathematician but was not allowed to
actual matrices. James Joseph Sylvester first used graduate because, as a Jew,
The first verified instance of the the term “matrix” to describe an he would not swear allegiance
to the Church of England.
use of matrices comes from ancient array of numbers. Shortly after
Sylvester taught briefly
China. In the second century bce, Sylvester introduced the term, in the US, but faced similar
the textbook The Nine Chapters his friend and colleague Arthur difficulties there. Returning to
on the Mathematical Art described Cayley formalized the rules for London, he studied law and
how to set out a counting board manipulating matrices. Cayley was admitted to the bar in
and use a matrix-like method to showed that the rules of matrix 1850. He also began to work
solve linear simultaneous equations algebra are different from those on matrices with fellow British
with several unknown values. in standard algebra. Two matrices mathematician Arthur Cayley.
This method was similar to the of the same size (with the same In 1876, Sylvester returned to
number of elements in their the US as a maths professor
respective rows and columns) at Johns Hopkins University,
are added by simply adding Maryland, where he founded
corresponding elements. Matrices the American Journal of
with different dimensions cannot Mathematics. Sylvester died
in London in 1897.
be added. Matrix multiplication
is, however, quite different from
multiplication of numbers. Not all ❯❯ Key works

1850 On a New Class of

The arrays found in Mayan relics Theorems
suggest to some historians that the 1852 On the principle of the
Maya used matrices to solve linear calculus of forms
equations. However, others believe 1876 Treatise on elliptic
they were merely replicating patterns functions
in nature, such as on a turtle’s shell.
240 MATRICES
Multiplying two matrices together is achieved by multiplying the horizontals determinants are not zero, form an
in the first matrix by the vertical numbers in the second (the centred dot indicates algebraic structure called a group.
multiplication) and adding the results. In matrix algebra, switching around the Theorems that are true for groups
order in which the two matrices are multiplied produces a different result, as
are therefore also true for such
shown here with the multiplication of two square matrices ( A and B).
matrices, and advances in group
A B theory can be applied to matrices.
Groups can also be represented
4 8 2 9 4  28  7 4  98  0 64 36
1 3  7 0  1  23  7 1  9  3  0  23 9 as matrices, enabling difficult
problems in group theory to be
expressed in terms of matrix
algebra, which is more easily solved.
B A Representation theory, as this field
2 9 4 8 2  49  1 2  89  3 17 43 is known, is applied in number
7 0  1 3  7  40  1 7  8  0  3  28 56 theory and analysis, and in physics.

Determinants
The determinant of a matrix was
matrices can be multiplied together; be repeatedly multiplied by itself. named by Gauss, due to the fact
in matrix multiplication, AB (see A square matrix of size n  n with that it determines whether the
above) can only be calculated if the value 1 along the diagonal system of equations represented
the row count of B is the same starting top left, and the value 0 by the matrix has a solution. As
as the column count of A. Matrix everywhere else, is called the long as the determinant is not zero,
multiplication is non-commutative, identity matrix ( In). the system will have a unique
meaning that even where both A Every square matrix has an solution. If the determinant is
and B are square matrices, AB is associated value called its zero, the system may have either
not equal to BA. determinant, which encodes many no solution or many.
of the matrix’s properties and can be In the 17th century, Japanese
Square matrices computed by arithmetic operations mathematician Seki Takakaze
Because of their symmetry, square on the matrix’s elements. Square had shown how to calculate the
matrices have particular properties. matrices whose elements are determinants of matrices up to size
For example, a square matrix can complex numbers, and whose 5  5. Over the following century,

A linear transformation in 2 dimensions maps lines distance from the fixed line). The image of any point ( x, y)
through the origin to other lines through the origin, and is found by multiplying the matrix by the column vector
parallel lines to parallel lines. Linear transformations include representing the point ( x, y). In the examples below, the
rotations, reflections, enlargements, stretches, and shears original shape is the pink square, with vertices (0, 0), (2, 0),
(lines that slide parallel to a fixed line, in proportion to their (2, 2) and (0, 2), and the image is the green quadrilateral.
Horizontal shear with shear factor 1 Reflection in the vertical axis Enlargement by factor 1.5

1 1 x -1 0 x 1.5 0 x
0 1  y 0 1  y 0 1.5  y
y
y
y

x x x
 THE 19TH CENTURY 241

Matrices can store a large number of elements

in a compact, elegant form.

In control theory, a
Computers process transfer matrix can be
Banks use matrices
numbers stored in used to relate the input
for encryption.
huge matrices. and output of an
electronic system.

Matrices underpin much of the

technology we use today.

mathematicians uncovered the are encoded by 2  2 matrices, in computer science and

rules for finding determinants of while 3-D transformations involve cryptography. Stochastic matrices,
larger and larger arrays. In 1750, 3  3 matrices. The determinant whose elements represent
Swiss mathematician Gabriel of a transformation matrix contains probabilities, are used by search
Cramer stated a general rule information about the area or engine algorithms for ranking web
(now called Cramer’s rule) for the volume of the transformed figure. pages. Programmers use matrices
determinant of a matrix with m Today, computer aided design as keys when encrypting messages;
rows and n columns, but he failed (CAD) software makes extensive letters are assigned individual
to give the proof of this rule. use of matrices for this purpose. numerical values, which are then
In 1812, French mathematicians multiplied by the numbers in the
Augustin-Louis Cauchy and Modern applications matrix. The larger the matrix used,
Jacques Binet proved that when Matrices can store vast amounts the more secure the encryption is. ■
two square matrices of the same of data compactly, making them
size are multiplied, the determinant essential across maths, physics,
of this product is, in fact, the same and computing. Graph theory uses
as the product of their individual matrices to encode how a set of
determinants: detAB = (detA)  vertices (points) is connected by
(detB). This rule simplified the edges (lines). One formulation of
process of finding the determinant quantum physics, called matrix I have not thought it necessary
of a very large matrix by breaking it mechanics, makes extensive use to undertake the labour
down into the determinants of two of matrix algebra, and particle of a formal proof of the
smaller matrices. physicists and cosmologists use theorem in the general case
transformation matrices and group of a matrix of any degree.
Transformation matrices theory to study the symmetries Arthur Cayley
Matrices can be used to represent of the Universe.
linear geometric transformations Matrices are used to represent
(see opposite) such as reflections, electrical circuits for solving
rotations, translations, and scalings. problems about voltage and
Transformations in two dimensions current. They are also important
AN INVESTIGATION INTO
THE LAWS OF
THOUGHT
BOOLEAN ALGEBRA
244 BOOLEAN ALGEBRA

IN CONTEXT Boolean logic stipulates that outcomes

KEY FIGURE of all operations in Boolean algebra can
George Boole (1815–64) only be either true or false.

FIELD
Logic
BEFORE
350 bce Aristotle’s philosophy All operations in
Boolean algebra have “True” is usually
discusses syllogisms. represented by 1, and
only two possible “false” by 0.
1697 Gottfried Leibniz tries, outcomes: 1 or 0.
unsuccessfully, to use algebra
to formalize logic.
AFTER

L
1881 John Venn introduces ogic is the bedrock of a symbolic logic where arguments
Venn diagrams to explain mathematics. It provides us could be expressed using abstract
Boolean logic. with the rules of reasoning symbols. One of the pioneers of this
and gives us a basis for deciding shift to mathematical logic was
1893 Charles Sanders Peirce on the validity of an argument or British mathematician George
uses truth tables to show proposition. A mathematical Boole, who sought to apply methods
outcomes of Boolean algebra. argument uses the rules of logic from the emerging field of symbolic
to ensure that if a basic proposition algebra to logic.
1937 Claude Shannon uses
is true, then any and all statements
Boolean logic as the basis
constructed from that proposition Algebraic logic
for computer design in his will also be true. Boole’s investigations into logic
A Symbolic Analysis of Relay
The earliest attempt to set out began in an unconventional way.
and Switching Circuits.
the principles of logic was carried In 1847, a friend, British logician
out by the Greek philosopher Augustus De Morgan, became
Aristotle around 350 bce. His involved in a dispute with a
analysis of the various forms of philosopher about who deserved
arguments marked the beginning the credit for a particular idea.
of logic as a subject for study in Boole was not directly involved,
its own right. In particular, but the event spurred him to
Aristotle looked at a type of set down his ideas concerning
argument known as a syllogism, how logic could be formalized
consisting of three propositions. with mathematics, in his 1847
Mathematics had never more The first two propositions, called essay Mathematical Analysis
than a secondary interest for the premises, logically entail the of Logic.
him, and even logic he cared third proposition, the conclusion. Boole wanted to discover a
for chiefly as a means of Aristotle’s ideas about logic were way to frame logical arguments
clearing the ground. unrivalled and unchallenged in so that they could be manipulated
Mary Everest Boole Western thought for more than and solved mathematically. In
British mathematician and wife 2,000 years. order to achieve this, he developed
of George Boole Aristotle approached logic as a type of linguistic algebra, in
a branch of philosophy, but in the which the operations of ordinary
1800s, scholars began to study algebra, such as addition and
logic as a mathematical discipline. multiplication, were replaced
This involved moving from by the connectors that were used
arguments expressed in words to in logic. As in algebra, Boole’s
 THE 19TH CENTURY 245
See also: Syllogistic logic 50–51 ■ Binary numbers 176–77 ■ The algebraic resolution of equations 200–01 ■ Venn
diagrams 254 ■ The Turing machine 284–89 ■ Information theory 291 ■ Fuzzy logic 300–01

use of symbols and connectives of numbers and realized that the

allowed for the simplification of set {0, 1}, together with operations
logical expressions. such as addition and multiplication,
The three key operations of could be used to form a consistent
Boole’s algebra were and, or, and algebraic language. Boole proposed
not; Boole believed these were that logical propositions could Boolean algebra makes it
the only operations necessary have only two values – true or possible to prove logical
to perform comparisons of sets, false – and could not be anything statements by performing
as well as basic mathematical in between. algebraic calculations.
functions. For example, in logic, two In Boole’s logical algebra, truth Ian Stewart
statements may be connected by and falsity were reduced to binary British mathematician
and, as in “this animal is covered values: 1 for true and 0 for false.
in hair” and “this animal feeds its Starting out with an initial
young with milk”, or by or, as in statement that was either true or
“this animal can swim” or “this false, Boole could then construct
animal has feathers”. The statement further statements and use the
“A and B” is true when A and B are and, or, and not operations in
both individually true, whereas the order to determine whether or not algebra, there are only two possible
statement “A or B” is true if one or these further statements were true. values for any quantity, either 1
both of A and B is true. In Boolean or 0. There is also no such thing
terms, such statements can be One plus one is one as subtraction in Boole’s algebra.
given as, for example: (A or B) = Despite the resemblance, Boole’s For example, if statement A,
(B or A); not (not A) = A; or even true and false binary of 1 and 0 is “my dog is hairy”, is true, it has
not (A or B) = (not A) and (not B). not the same as binary numbers. a value of 1, and if statement B,
Boolean numbers are entirely “my dog is brown” is true, it also
Boole’s binaries different from the mathematics has a value of 1. A and B can be
In 1854, Boole published his most of real numbers. The “laws” of combined to make the statement
important work, An investigation Boole’s algebra allow statements “my dog is hairy or my dog is
into the laws of thought. Boole had that would not be permitted by brown”, which is also true, and
studied the algebraic properties other forms of algebra. In Boole’s also has a value of 1. In Boolean ❯❯

George Boole Born in Lincoln in 1815, George professor of mathematics at

Boole was the son of a shoemaker the new Queen’s College in
who passed his love of science Cork, Ireland, where he
and mathematics on to him. When remained until his premature
his father’s business collapsed, death at the age of 49.
the 16-year-old George took up a
post as an assistant schoolmaster
to support his family. He began Key works
to study mathematics seriously,
starting by reading a book on 1847 Mathematical Analysis
calculus. He later published work of Logic
in the Cambridge Mathematical 1854 An investigation into the
Journal, but still could not afford laws of thought
to study for a degree. 1859 Treatise on differential
In 1849, as a result of his equations
correspondence with Augustus 1860 Treatise on the calculus of
De Morgan, Boole was appointed finite differences
246 BOOLEAN ALGEBRA
depict relations of inclusion (and) statement is true, as it will only
and exclusion (not) between sets. be false if both A and B are false.
They consist of intersecting circles, More complex statements can
each one representing a distinct also be assessed by drawing truth
set. A two-circle Venn diagram tables. For example, A and ( B or
The furthest thing from my represents propositions such as: not C) is true when A and B are
mind has been those efforts “All A are B”, while a three-circle both true and C is false, and is false
which try to establish an diagram represents propositions when A is false and both B and C
artificial similarity [between involving three sets (such as X, Y, are true. Out of eight possible
logic and algebra]. and Z below). combinations of true and false,
Gottlob Frege The results of a statement in there are three in which the
Boolean algebra can also be statement is true and five in
assessed using a truth table (see which it is false.
opposite), in which all possible
input combinations are tried and Limitations
written out. These truth tables One drawback in Boole’s system
were first used by American of algebra was that it contained
algebra, or behaves like + (apart logician Charles Saunders Peirce in no method of quantification: there
from 1 + 1 = 1) and and behaves 1893, nearly 30 years after Boole’s was no simple way of expressing
like  (see table on p.247). death. For example, the statement a statement such as “for all X ”, for
A and B can only be considered example. The first symbolic logic
Visualizing results true if both A and B are true. If with quantification was produced
One way of visualizing Boole’s one or both of A and B are false, in 1879 by German logician Gottlob
algebra is in the form of diagrams then A and B is false. Therefore, out Frege, who objected to Boole’s
invented by British logician John of the four possible combinations of attempts to turn logic into algebra.
Venn. In his work Symbolic Logic A and B, only one results in a true Frege’s work was followed by
(1881), Venn developed Boole’s answer. On the other hand, for Charles Sanders Peirce and
theories employing what became A or B, there are three possible another German logician, Ernst
known as Venn diagrams. These combinations in which that Schröder, who introduced

These Venn diagrams represent

three of the most basic functions in
Boolean algebra: the functions for and,
or, and not. The three-circle diagram
represents a combination of two
functions: ( X and Y ) or Z.

The region showing the X Y Z

output of the function X and Y

X Y X Y X Y
X or Y X not Y (X and Y ) or Z
 THE 19TH CENTURY 247

Gate Symbol Truth table

NOT INPUT OUTPUT

A NOT gate’s A X 1 0
output is the opposite
of its input. 0 1

INPUT OUTPUT
AND A B A AND B
This logic module is used for An AND gate’s A 0 0 0
X
teaching how logic gates function in output is 1 only if B 0 1 0
electronic circuits. The gates can be both its inputs are 1.
1 0 0
connected to lights or buzzers which
go on and off depending on the output. 1 1 1

INPUT OUTPUT
quantification into Boole’s algebra A B A AND B
OR
and produced substantial works A 0 0 0
An OR gate’s output X
using Boole’s system. is 0 only if both its B 0 1 1
inputs are 0.
1 0 1
Boole’s legacy 1 1 1
It was not until some 70 years
after Boole’s death that the
potential of his ideas was fully INPUT OUTPUT
grasped. American engineer NAND A B A AND B
Claude Shannon used Boole’s A NAND gate is an A 0 0 1
X
Mathematical Analysis of Logic AND gate followed by B 0 1 1
to establish the basis of modern a NOT gate.
1 0 1
digital computer circuits. 1 1 0
While working on the electrical
circuitry for one of the world’s first
INPUT OUTPUT
computers, Shannon realized that
Boole’s two-value binary system NOR A B A AND B
could be the basis of logic gates A NOR gate is an A 0 0 1
X
(physical devices that move based OR gate followed by B 0 1 0
a NOT gate.
on Boolean functions) in the 1 0 0
circuitry. Aged just 21, Shannon 1 1 0
published the ideas that would
form the basis of future computer Logic gates, which are physical electronic devices implementing
design in A Symbolic Analysis of Boolean functions, form an important part of computer circuitry. This
Relay and Switching Circuits, table shows the various symbols for each type of logic gate. Truth tables
published in 1937. show the possible outcomes of various inputs into the gate.
The building blocks of codes
now used to program computer
software are based on the logic commonly used to filter results have simply become silent: a search
formulated by Boole. Boolean to find the specific thing being for “George Boole”, for example,
logic is also at the heart of how searched for, but advances in has an implied and between the
internet search engines work. In technology allow people today two words, so that only web pages
the early days of the internet, the to search using more natural containing both names will appear
and, or, and not commands were language. The Boolean commands in the results. ■
248

A SHAPE WITH
JUST ONE SIDE
THE MÖBIUS STRIP

IN CONTEXT 1. Twist a
strip of paper
KEY FIGURE through 180°.
August Möbius (1790–1868)
FIELD 2. Join the ends of
Applied geometry the strip of paper.

BEFORE
3rd century ce A Roman
mosaic of Aion, Greek god of
eternal time, features a zodiac 3. The strip
shaped like a Möbius strip. now has a
single surface.
1847 Johann Listing
publishes Vorstudien zur A Möbius strip can be made from a simple length of paper. It can be
Topologie (Introductory coloured in with a crayon in one continuous movement, without taking
Studies in Topology). the crayon away from the paper. The shape has a single surface; this
can be tested by following the surface of the shape with the eye.
AFTER

N
1882 Felix Klein describes
the Kleinsche Flasche (Klein amed after 19th-century spawned many new geometrical
bottle), a shape composed of German mathematician shapes. Much of this impetus came
August Möbius, a Möbius from German mathematicians,
two Möbius strips.
strip can be created in seconds by including Möbius and Johann
1957 In the US, the B. F. twisting a strip of paper through Listing. In 1858, the two men
Goodrich Company produces 180°, then joining its two ends independently investigated the
a patent for a conveyor belt together. The shape that results has twisted strip, which Listing is
based on the Möbius strip. some unexpected properties, which said to have discovered first.
have advanced our understanding Once formed, the Möbius
2015 Möbius strips are used of complex geometrical figures – a strip has only one surface – an ant
in laser beam research, branch of study called topology. crawling along that surface would
with potential application The 19th century was a creative be able to cover both sides of the
in nanotechnology. period for mathematics, and the paper in one continuous movement
exciting new field of topology without crossing the edge of the
 THE 19TH CENTURY 249
See also: Graph theory 194–95 ■ Topology 256–59 ■ Minkowski space 274–75
■Fractals 306–11

A Roman mosaic dating from

c. 200 ce includes what may be the
earliest representation of a Möbius
strip, which is thought to represent
the eternal nature of time.

and in the molecular structure of

some proteins. Its properties have
been put to use in everyday
applications, too. In the early
20th century, the Möbius strip August Möbius
shape was used in continuous-loop
recording tapes to provide double Born near Naumberg, in
the playback time. There are also Saxony, Germany, in 1790,
Möbius strip roller-coasters, such August Ferdinand Möbius
as the Grand National at Blackpool was the son of a dance
Pleasure Beach in northern England. teacher. At the age of 18,
paper. In geometry, it is considered The Möbius strip’s form has he entered the University of
a classic example of a “non- inspired artists and architects. Leipzig to study mathematics,
orientable” surface. This means Dutch artist Maurits Escher physics, and astronomy, and
that when you trace your finger created a notable woodcut of ants later studied in Göttingen
around the complete strip, the left endlessly patrolling the shape. under the great German
mathematician Carl Friedrich
and right sides of the paper are Impressive Möbius strip buildings
Gauss. In 1816, Möbius was
reversed. The Möbius strip is the are being constructed to minimize
appointed professor of
simplest non-orientable, two- the impact of the sun’s rays. The astronomy at Leipzig and
dimensional surface that can be shape is used in the universal stayed there for the rest of
created in three-dimensional space. symbol for recycling, and also his life, writing treatises on
Experimenting with the Möbius suggested in the mathematical Halley’s Comet and other
strip produces other unexpected symbol for infinity (), echoing aspects of astronomy.
results. For instance, if you draw a the eternity image in the ancient Möbius is associated with
line around the centre of the strip Roman mosaic (above). ■ a number of mathematical
and then cut along it, the shape concepts, including Möbius
does not divide in half. Rather, it transformations, the Möbius
produces a longer, continuous function, the Möbius plane,
twisted loop. Alternatively, draw a and the Mobius inversion
line about a third of the way across formula. He also conjectured a
the width of the strip then turn the geometrical projection known
Our lives are Möbius strips, as a Möbius net. Möbius died
scissors 90° and cut along its misery and wonder in Leipzig in 1868.
length: the result is one twisted simultaneously. Our
loop linked to a second, thinner destinies are infinite,
twisted loop that is twice as long.
and infinitely recurring. Key works
Space, industry, and art
Joyce Carol Oates
American novelist 1827 The Calculus of Centres
The Möbius strip shape sometimes
of Gravity
occurs naturally, such as in the 1837 Textbook of Statics
movement of magnetically charged 1843 The Elements of Celestial
particles within the Van Allen Mechanics
radiation belts that surround Earth,
250

THE MUSIC
OF THE PRIMES
THE RIEMANN HYPOTHESIS

IN CONTEXT
It is very difficult to estimate how many prime numbers
KEY FIGURE there are between a pair of numbers.
Bernhard Riemann
(1826–66)
FIELD
Number theory The Riemann hypothesis states that the zeta function
BEFORE (a function in number theory) gives the most accurate estimate
for the number of primes between two values.
1748 Leonhard Euler defines
the Euler product, linking a
version of what will become
the zeta function to the
sequence of prime numbers.
The hypothesis has not yet been proven.
1848 Russian mathematician
Pafnuty Chebyshev presents
the first significant study of the

I
prime counting function (n). n 1900, David Hilbert listed between 1 and 100, 25 are prime (1 in
AFTER 23 outstanding mathematical 4); between 1 and 100,000, 9,592 are
1901 Swedish mathematician problems. One of them was prime (about 1 in 10). These values
the Riemann hypothesis, which are expressed through the prime
Helge von Koch proves that
is still agreed to be one of the most counting function, (n), but  here
the best possible version of the
important unsolved problems in is not related to the mathematical
prime counting function relies
mathematics. It concerns the prime constant pi. Inputting n into 
on the Riemann hypothesis. numbers – numbers that are only gives the number of primes between
2004 Distributed computing divisible by themselves or 1. Proving 1 and n. For example, the number of
is used to prove that the first the Riemann hypothesis would solve primes up to 100 gives (100) = 25.
10 trillion “non-trivial zeros” many other theorems.
lie on the critical line. The most noticeable thing about Finding the pattern
prime numbers is that the larger For centuries, mathematicians’
they are, the more widely spread fascination with primes has led
out they get. Of the numbers them to seek a formula that would
 THE 19TH CENTURY 251
See also: ■ Mersenne primes 124 ■ Imaginary and complex numbers 128–31
■ The complex plane 214–15 ■ The prime number theorem 260–61

the complex plane, they all lie on

“the critical line”, where the real part
of the number is 0.5. This belief is
called the Riemann hypothesis.

The failure of the Riemann A solution

hypothesis would create In 2018, British mathematician
havoc in the distribution Michael Atiyah, then aged 89, said
of prime numbers. he had found a simple proof for the
Enrico Bombieri Riemann hypothesis. He died a few
Italian mathematician months later, the proof unverified. Bernhard Riemann
Although proving the Riemann
hypothesis would validate the zeta The son of a pastor, Bernhard
function's status as the best Riemann was born in
predictor of the distribution of Germany in 1826. Initially
primes, it still would not allow prime fascinated by theology, he
numbers to be fully predicted. was persuaded to change his
predict the values of this function. Their distribution is to some extent degree to mathematics by Carl
Aged just 14, Carl Gauss found a chaotic. But the hypothesis does pin Gauss, under whom he then
rough answer, and he was soon down the blend of predictability and studied at the University of
able to find an improved version of randomness the primes obey. This Göttingen. The result was a
the prime counting function that blend is exactly that exhibited by series of breakthroughs that
remain influential today.
could predict the number of primes the energy levels of the nuclei of
In addition to his work on
between 1 and 1,000,000 as 78,628, heavy atoms, according to quantum
primes, Riemann helped to
which is accurate to 0.2 per cent. theory. This profound connection formulate the rules for applying
means the hypothesis may one day calculus to complex functions
A new formula be proved not by a mathematician, (functions using complex
In 1859, Bernhard Riemann but by a physicist. ■ numbers). His revolutionary
constructed a new formula for understanding of space was
(n), which would give the most used by Einstein in developing
accurate estimates possible. One of relativity theory. Despite his
the inputs needed for this formula success, Riemann struggled
is a series of complex numbers financially. He could finally
defined by what is now called the afford to marry when he was
Riemann zeta function, z(s). awarded a full professorship
The numbers that are needed to by Göttingen in 1862. Just a
confirm Riemann’s formula for (n) month later, he fell ill and his
health deteriorated until he
are those complex numbers (s) for
died of tuberculosis in 1866.
which z(s) = 0. Some of these – the
“trivial zeros” – are easy to find; they
are all the negative even integers
(-2, -4, -6, and so on). Finding Key work
the others (the “non-trivial zeros” –
1868 Ueber die Hypothesen,
all other values for which z(s) = 0)
The uranium atom is one example welche der Geometrie zu
is more difficult. Riemann only of a heavy atom whose nucleus follows Grunde liegen (On the
calculated three. He believed that the same statistical behaviour as Hypotheses Which Lie at the
non-trivial zeros have one thing in prime numbers, making it extremely Foundation of Geometry)
common: when they are plotted on difficult to predict.
252

SOME INFINITIES
ARE BIGGER
THAN OTHERS
TRANSFINITE NUMBERS

I
nfinity was a concept that than others. In order to describe
IN CONTEXT mathematicians had long these differing infinities, he
instinctively mistrusted. It introduced “transfinite” numbers.
KEY FIGURE
was only in the late 19th century While he was studying set
Georg Cantor (1845–1918)
that Georg Cantor was able to theory, Cantor aimed to create
FIELD explain it with mathematical rigour. definitions for every number to
Number theory He found there was more than one infinity. This need arose from
kind of infinity – an infinite variety, the discovery of transcendental
BEFORE in fact – and that some were larger numbers, such as p and e, which
450 bce Zeno of Elea uses a
series of paradoxes to explore
the nature of infinity.
The infinite set of natural The infinite set of
1844 French mathematician numbers (positive integers) transcendental numbers,
Joseph Liouville proves is well ordered and such as p, cannot be listed
that a number can be theoretically can be listed. in any order.
transcendental – have an
infinite number of digits
arranged with no repeating
pattern and without an
algebraic root. So it is a So it is an
countable infinity. uncountable infinity.
AFTER
1901 Bertrand Russell’s
barber paradox exposes the
weakness of set theory’s
ability to define numbers. An uncountable infinity is larger than
a countable infinity.
1913 The infinite monkey
theorem explains that given
infinite time, random input
will eventually produce all
possible outcomes. Some infinities are bigger than others.
 THE 19TH CENTURY 253
See also: Irrational numbers 44–45 ■ Zeno’s paradoxes of motion 46–47 ■ Negative numbers 76–79 ■ Imaginary and
complex numbers 128–31 ■ Calculus 168–75 ■ The logic of mathematics 272–73 ■ The infinite monkey theorem 278–79

are irrational, infinitely long, and Every number within this diagram is
real, as opposed to imaginary: it gives
are not themselves an algebraic a positive result when it is squared.
A transcendental
root. Between every algebraic number can
Real never be fully
number – including the integers, calculated and
Real algebraic
fractions, and certain irrational so cannot be
Rational  added to a set of
numbers (such as √2) – there is an
Integer numbers in the
infinite number of transcendentals. 2 e correct order,
1 1/3 thus forming an
Counting infinities 3 2 uncountable set.
Natural 2
To help identify where a number is 2/3 15/2
1 2 3
located, Cantor drew a distinction 3 Transcendental
between two kinds of numbers: 2.25 The numbers in these
cardinals, which are the counting two bands are irrational
Irrational
numbers 1, 2, 3… that denote the as they cannot be
size of a set; and ordinals, such as described as fractions
containing two integers.
1st, 2nd, or 3rd, which list order.
Cantor created a new transfinite These concentric rings show the different types of numbers,
cardinal number – aleph (), the which correspond to different types of infinities. Each ring
first letter of the Hebrew alphabet – describes a set of numbers. For example, the set of natural numbers
to denote a set containing an infinite is a small subset of rational numbers, which in turn combine with
the set of irrational numbers to make the full set of real numbers.
number of elements. The set of
integers that includes the natural
numbers, negative integers, and item, a transfinite ordinal number. making this infinity larger than
zero, was given the cardinality of The number of items in a set with countable ones, so it is said to
0, the smallest transfinite cardinal, a cardinality of 0 is . have a cardinality of 1.
as these are theoretically countable Adding to that set makes a new The set of all 1 sets contains
numbers but are actually impossible set of  + 1. A set of all countable 2 items, with a cardinality of 2.
to count completely. A set with a ordinals, such as  + 1,  + 1 + 2, In this way, Cantor’s set theory
cardinality of 0 starts with a first  + 1 + 2 + 3…, will contain 1 creates infinities nestled inside
item, and ends with a  (omega) items. This set cannot be counted, each, expanding forever. ■

Georg Cantor Born in St Petersburg, Russia, in criticized by the clergy, but

1845, Georg Cantor moved with Cantor, who was deeply
his family to Germany in 1856. religious, saw his research
An outstanding scholar (and as a glorification of God.
violinist), he studied in Berlin Overwhelmed by depression,
and Göttingen. He was later Cantor was institutionalized for
made a professor of mathematics much of his later life. He began
at the University of Halle. to receive plaudits in the early
Although much admired by 20th century, but lived out his
today’s mathematicians, Cantor old age in poverty. He died of
was something of a pariah among a heart attack in 1918.
his contemporaries. His theory of
transfinite numbers clashed with Key work
traditional mathematical beliefs
and the criticisms of leading 1915 Contributions to the
mathematicians damaged his founding of the theory of
career. His work was also transfinite numbers
254

A DIAGRAMMATIC
REPRESENTATION
OF REASONINGS
VENN DIAGRAMS

I
n 1880, British mathematician and a categorical conclusion. For
IN CONTEXT John Venn introduced the example: “All French people are
idea of the Venn diagram in European. Some French people eat
KEY FIGURE
his paper “On the Diagrammatic cheese. Therefore, some Europeans
John Venn (1834–1923)
and Mechanical Representation of eat cheese.”
FIELD Propositions and Reasonings”. The As well as being a widely
Statistics Venn diagram is a way of grouping used tool for sorting data in
things in overlapping circles (or everyday life, in contexts ranging
BEFORE other curved shapes) to show the from school classrooms to
c. 1290 Catalan mystic Ramon relationship between them. boardrooms, Venn diagrams are
Llull devises classification an integral part of set theory,
systems using devices such Overlapping circles due to their distinctive ability
as trees, ladders, and wheels. The Venn diagram considers two to express relationships. ■
or three different sets or groups of
c. 1690 Gottfried Leibniz things with something in common,
creates classification circles. such as all living things, or all
1762 Leonhard Euler planets of the solar system. Each
describes the use of logic set is given its own circle and the
circles, now known as circles are overlapped. Objects in Great ideas are the ones
“Euler circles”. each set are then arranged in the that lie in the intersection
circles so that objects that belong of the Venn diagram of
AFTER in more than one set are placed ‘is a good idea’ and
1963 American mathematician where the circles overlap.
‘looks like a bad idea’.
David W. Henderson outlines Two-circle Venn diagrams can
represent categorical propositions,
Sam Altman
the connection between American entrepreneur
symmetrical Venn diagrams such as “All A are B”, “No A are B”,
and prime numbers. “Some A are B”, and “Some A are
not B”. Three-circle diagrams can
2003 In the US, Jerrold also represent syllogisms, in which
Griggs, Charles Killian, there are two categorical premises
and Carla Savage show that
symmetrical Venn diagrams See also: Syllogistic logic 50–51 ■ Probability 162–65 ■ Calculus 168–75
exist for all primes. ■ Euler’s number 186–91 ■ The logic of mathematics 272–73
 THE 19TH CENTURY 255

THE TOWER WILL

FALL AND THE
WORLD WILL
THE TOWER OF HANOI
END

F
rench mathematician
IN CONTEXT Édouard Lucas is believed
to have invented his Tower
KEY FIGURE
of Hanoi game in 1883. The aim of
Édouard Lucas (1842–91)
the puzzle is simple. The challenger
FIELD is presented with three poles,
Number theory one of which holds three discs in
order of size, with the largest
BEFORE disc on the bottom. The three discs
1876 Édouard Lucas proves must be moved one disc at a time
that the Mersenne number so as to recreate the starting
2127  1 is prime. This is still arrangement on a different pole
the largest prime ever found A form of the Tower of Hanoi is a
using the smallest possible number
popular toy for small children. Versions
without using a computer. of moves, with the restriction that with eight discs are often used to test
players can only place a disc on developmental skills of older children.
AFTER top of a larger disc or onto an
1894 Lucas’s work on empty pole.
recreational mathematics starting pole; 1 shows that it is on
is posthumously published Solving the puzzle the final pole. The sequence of bits
in four volumes. With just three discs, the Tower of changes at each move.
1959 American writer Erik Hanoi can be solved in just seven According to legend, if monks at
Frank Russell publishes “Now moves. With any number of discs, a certain temple in either India or
n
the formula 2  1 will give the Vietnam (depending on the version
Inhale”, a short story about
minimum number of moves (where of the tale) succeed in moving 64
an alien allowed to play a n is equal to the number of discs). discs from one pole to another in
version of the Tower of Hanoi One solution to the challenge line with the rules, the world will
before his execution. employs binary numbers (0 and 1). end. However, even using the best
1966 In an episode of the Each disc is represented by a strategy and moving one disc per
BBC’s Doctor Who, the villain, binary digit, or bit. A value of 0 second, they would take 585 billion
The Celestial Toymaker, forces indicates that a disc is on the years to complete the game. ■
the Doctor to play a ten-disc
version of the game. See also: Wheat on a chessboard 112–13 ■ Mersenne primes 124 ■ Binary
numbers 176–77
256
IN CONTEXT

SIZE AND
KEY FIGURE
Henri Poincaré (1854–1912)

SHAPE DO NOT
FIELD
Geometry
BEFORE
1736 Leonhard Euler solves

MATTER, ONLY
the historical topological
problem of “The Seven Bridges
of Königsberg”.

CONNECTIONS
1847 Johann Listing coins
the term “topology” as a
mathematical subject.
AFTER
TOPOLOGY 1925 Russian mathematician
Pavel Aleksandrov establishes
the basis for studying the
essential properties of
topological spaces.
2006 Grigori Perelman’s proof
of the Poincaré conjecture is
confirmed.

T
opology is, in simple terms,
the study of shapes without
measurements. In classical
geometry, if a pair of shapes has
equal corresponding lengths and
angles, and you can slide, reflect,
or rotate one of the shapes into
the other, they are congruent –
a mathematical way of saying
they are identical. To a topologist,
however, two shapes are identical –
or invariant, in topological
terminology – if they can be
moulded one into the other by
continuous stretching, twisting,
or bending, but with no cutting,
piercing, or sticking together. This
has led to topology being called
“rubber-sheet geometry”.
 THE 19TH CENTURY 257
See also: Euclid’s Elements 52–57 ■ Coordinates 144–51 ■ The Möbius strip
248–49 ■ Minkowski space 274–75 ■ Proving the Poincaré conjecture 324–25

Topology is the study of abstract shapes

without measurements.

Topologically identical shapes can be moulded into each

other through stretching, twisting, or bending.
Henri Poincaré
Born in 1854, in Nancy, France,
Henri Poincaré showed such
early promise that he was
described by a teacher as a
“monster of mathematics”.
Shape and size do not matter, He graduated in the subject
only connections (the number of holes). from the Paris École
Polytechnique and earned his
doctorate from the University
of Paris. In 1886, he was
appointed as chair of
mathematical physics and
For more than 2,000 years, from the dimensional figures with four or
probability at the Sorbonne
time of Euclid, c. 300 bce, geometry more planes, such as a cube or in Paris, where he spent the
was concerned with classifying pyramid – that involved their rest of his career.
shapes by their lengths and angles. vertices, edges, and faces rather In 1887, Poincaré won a
In the 18th and early 19th centuries, than lines and angles. What he prize from King Oscar II of
some mathematicians began to postulated became known as Sweden for his partial solution
look at geometric objects differently, Euler’s polyhedral formula: ❯❯ of the many variables involved
considering the global properties in determining the stable orbit
of shapes beyond the confines of of three planets around one
Face ( )
F
lines and angles. Out of this grew another. A self-confessed
the mathematical field of topology, mistake threw his calculations
which by the early 20th century for the stable orbit into doubt,
had moved far from the notion of but in turn paved the way for
“shape” to embrace abstract the study of “chaos theory”.
He died in 1912.
algebraic structures. The most
ambitious and influential exponent
of this was French mathematician Key works
Henri Poincaré, who used complex Edge ( E)
topology to throw new light on the Vertex ( )
V 1892–99 Les Méthodes
“shape” of the Universe itself. nouvelles de la mécanique
Euler’s formula, V + F - E = 2, céleste (New Methods of
works for most polyhedra, including a Celestial Mechanics)
Birth of a new geometry cube. Its values of V = 8, F = 6, and 1895 Analysis Situs (Topology)
In 1750, Leonhard Euler revealed E = 12, when fed into the formula, 1903 La Science et l’hypothèse
that he had been working on a produce the calculation 8 + 6 - 12 (Science and Hypothesis)
formula for polyhedra – three- which equals 2.
258 TOPOLOGY
related regardless of how much only one surface, is non-orientable
they might be manipulated. (has no “left” or “right”), and, unlike
The term “topology” – derived a Möbius strip, has no edge or
from the Greek topos, meaning boundary curve. As it has no
“a place” – was introduced to intersections, the shape can only
Algebraic topology allows the mathematical world by German truly exist in four-dimensional
us to read qualitative mathematician Johann Listing in space. If the shape is represented
forms and their 1847 in his treatise Vorstudien zur in 3-D, it has to intersect itself,
transformations. Topologie (Introductory Studies which is where it starts to look like
Stephanie Strickland in Topology), although he had used a bottle. Topologists applied the
American poet the word in correspondence at least term “2-manifold” to shapes such as
10 years earlier. In particular, the Möbius strip and Klein bottle to
Listing was interested in shapes describe their surfaces, which
that did not satisfy Euler’s formula are two-dimensional surfaces
or defied the conventions of having embedded within a space of higher
distinct “outside” and “inside” dimension (the Möbius strip can
surfaces. He even devised a version exist inside three dimensional
V + F - E = 2, where V is the of the Möbius strip – a surface that space, but the Klein bottle can
number of vertices, F the number of has only one side and one edge – a only exist properly in four).
faces, and E the number of edges. few months before August Möbius.
The formula suggested that all Around the same period, A universal conjecture
polyhedra shared basic another German mathematician, The shape of the Universe has long
characteristics. Bernhard Riemann, devised new been a source of speculation. We
However, in 1813, another Swiss geometrical coordinate systems appear to inhabit a 3-D world, but
mathematician, Simone L’Huilier, that extended beyond the limits of to make any sense of its shape we
noted that Euler’s formula was not the 2-D and 3-D systems devised need to take ourselves outside this,
true for all polyhedra; it was false by René Descartes. Riemann’s new into four dimensions. In the same
for polyhedra with holes and for framework enabled mathematicians way, to gain a sense of the shape
non-convex polyhedra – shapes to explore shapes in four dimensions of a 2-D surface, we need to look
with some diagonals (linked by or higher, including seemingly down on it in three dimensions. A
vertices) not contained within or “impossible” shapes. starting point would be to imagine
on the surface. L’Huilier devised a One such shape was the “Klein that we inhabit a Universe that is a
system whereby every shape had bottle”, devised in 1882 by German 3-D surface embedded within four
its own “Euler characteristic” – mathematician Felix Klein. He dimensions. Taking this one step
(V - E + F ) – and shapes with the imagined joining two Möbius strips further, you could consider that this
same Euler characteristic were together to create a shape that has 3-D surface is actually a sphere

To a topologist, a coffee mug is identical in shape to a

doughnut, because by pulling, stretching, and bending one,
you could mould one into the shape of the other.

Coffee mug Doughnut

 THE 19TH CENTURY 259
The BlackDog™ robot is designed
to carry loads over rough terrain. The
robot’s moves are computed using
algebraic topology that can predict
and model the surrounding ”space”.

embedded in a 4-D space, also

known as a “3-sphere”. A “2-sphere”
is equivalent to a “normal” sphere
(such as a ball) in a 3-D space.
In 1904, Henri Poincaré went
even further, producing a theory
that would help to lay a topological
basis for understanding the shape
of the Universe. He proposed what
became known as the Poincaré
conjecture: “every simply connected,
closed 3-manifold is homeomorphic
to the 3-sphere”. A “3-manifold” is
a shape that appears 3-D when its
surface is enlarged but exists within
higher dimensions, and “simply
connected” means that it has no loop following the smooth contours coordinate system. This area of
holes – like an orange but not a of a sphere; if you had passed topological mathematics has
doughnut. A “closed” shape is finite, through holes or gaps, then the become known as “infinite-
with no boundaries – like a sphere. string could get “snagged”. For dimensional topology”.
Finally, “homeomorphic” describes example, if the Universe were The field of topology is now
shapes that can be moulded into shaped like a doughnut, and, in vast, embracing abstract algebraic
each other, such as a mug and a your travels, you wrapped your structures far removed from a
doughnut (see opposite). A doughnut string around its girth, the string simple notion of “shape”. It has
and an orange, however, are not would get caught. You would not be wide-ranging applications in areas
homeomorphic because of the hole able to gather in the string without such as genetics and molecular
in the doughnut. pulling it beyond the Universe. biology, such as helping to unravel
According to Poincaré, if it the “knots” created around DNA
could be could shown that the Shaping the future by certain enzymes. ■
Universe did not contain holes, then Topology developments still
you could model it as a “3-sphere”. continued during the 20th century.
To establish whether it contained In 1905, French mathematician
holes, you could, in theory, conduct Maurice Fréchet devised the idea
an experiment with string. Imagine of a metric space – a set of points
you are an explorer travelling around together with a “metric” that Probably no branch
the Universe from a set point, and defines the distance between them. of mathematics has
unravelling a ball of string as you Also at the turn of the 20th experienced a more
go. When you get back to your century, German mathematician
surprising growth.
starting point, you see the end of David Hilbert invented the idea of
the string that you started with. a space that took the Euclidean
Raymond Louis Wilder
American mathematician
You take both ends, and start to spaces of two and three dimensions
gather in the string, pulling both and generalized them to infinite
ends. If the Universe is “simply dimensions. Mathematics could
connected”, then you would be able then be done in any dimension in
to gather in the whole string, like a much the same way as in a 3-D
260

LOST IN
THAT SILENT,
MEASURED SPACE
THE PRIME NUMBER THEOREM

T
he prime numbers – those the distribution of primes. This
IN CONTEXT positive whole numbers became known as the prime
that have only two factors, number theorem. In 1896, Jacques
KEY FIGURE
themselves and 1 – have long Hadamard in France and Charles-
Jacques Hadamard
fascinated mathematicians. If the Jean de la Vallée Poussin in
(1865–1963)
first step was to find them, and Belgium both proved the theorem,
FIELD they are frequent among the small quite independently.
Number theory numbers, the next step was to It is evident that primes
identify a pattern to describe their decrease in frequency as numbers
BEFORE distribution. More than 2,000 years get larger. Of the first 20 positive
1798 French mathematician before, Euclid had proved that there whole numbers, eight are prime –
Adrien-Marie Legendre offers are infinitely many primes, but it 2, 3, 5, 7, 11, 13, 17, and 19. Between
an approximate formula to was only at the end of the 18th the numbers 1,000 and 1,020, there
determine how many prime century that Legendre stated his are only three prime numbers
numbers there are below or conjecture – a formula to describe (1,009, 1,013, 1,019), and between
equal to a given value.
1859 Bernhard Riemann
outlines a possible proof for the There are
There are There are
prime number theorem, but 25 prime 21 prime 16 prime
the necessary mathematics to numbers from numbers from numbers from
complete it does not yet exist. 1 to 100. 101 to 200. 201 to 300.
AFTER
1903 German mathematician
Edmund Landau simplifies
Hadamard’s proof of the prime
number theorem. Prime numbers become less common as numbers get larger.
1949 Paul Erdős in Hungary
and Atle Selberg in Norway
both find a proof of the theorem
using only number theory.
A pattern of primes emerges.
 THE 19TH CENTURY 261
See also: Euclid’s Elements 52–57 ■ Mersenne primes 124 ■ Imaginary and
complex numbers 128–31 ■ The Riemann hypothesis 250–51

1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50

Primes tend to decrease in frequency as numbers get Primes

larger. Although there are two primes between 30 and 40,
and three between 40 and 50, the accuracy of the prime Jacques Hadamard
number theorem increases at higher numbers.
Born in Versailles, France,
in 1865, Jacques-Salomon
1,000,000 and 1,000,020, the only natural logarithm of x. To explain Hadamard became interested
prime is 1,000,003. This seems the theorem slightly differently, for in mathematics thanks to an
reasonable; the higher the number, a large number x, the average gap inspiring teacher. He obtained
the more numbers that could be between primes from 1 to x is his doctorate in Paris in 1892
divisors exist below it. approximately ln( x). Or, for any and the same year won the
Many notable mathematicians number between 1 and x, the Grand Prix des Sciences
have puzzled over how primes are probability of it being a prime is Mathématiques for his work
distributed. In 1859, German approximately 1  ln( x). on primes. He moved to
Bordeaux to lecture at the
mathematician Bernhard Riemann The prime numbers are the
university, and there proved
worked towards a proof in his building blocks for numbers in
the prime number theorem.
paper On the Number of Primes mathematics, just as the elements In 1894, Alfred Dreyfus, a
Less Than a Given Magnitude. He are for compounds in chemistry. Jewish relative of Hadamard’s
believed that complex analysis, a Fundamental to understanding this wife, was falsely accused of
branch of mathematics in which is the Riemann hypothesis – an selling state secrets and was
ideas of function are applied to unsolved conjecture – which, if sentenced to life in prison.
complex numbers (combinations true, could reveal a huge amount Hadamard, who was also
of real numbers, such as 1, and more about prime numbers. ■ Jewish, worked tirelessly on
imaginary numbers, such as √-1), behalf of Dreyfus and he was
would lead to a resolution. He was eventually freed. Hadamard’s
right; the study of complex analysis brilliant career was marred
developed, fuelling the proofs of by personal loss; two of his
Hadamard and Poussin. sons died in World War I, and
another in World War II. The
The prime numbers… death of his grandson Étienne
What the theorem says grow like weeds among in 1962 was a final blow.
The prime number theorem is the natural numbers, Hadamard died a year later.
designed to calculate how many seeming to obey no other
primes there are less than or equal
law than that of chance. Key works
to a real number x. It states that
p( x) is approximately equal to
Don Zagier
American mathematician 1892 Determination of the
x  ln(x) as x gets larger and tends Number of Primes Less than
to infinity. Here p( x) denotes the a Given Number
prime counting function (how 1910 Lesson on the Calculus
many primes) and is unrelated to of Variations
the number pi, and ln( x) is the
MODERN
MATHEM
1900–PRESENT
ATICS
264 INTRODUCTION

David Hilbert defines Inspired by Einstein’s A group of French mathematicians

the 23 most important Bertrand Russell theory of special begin to write under the
unsolved problems in uses the barber relativity, Hermann pseudonym Nicolas Bourbaki,
mathematical research, paradox to demonstrate Minkowski suggests the with their work paving the way
setting the scene for contradictions in idea of spacetime as the for the eventual solution of
the century ahead. set theory. invisible fourth dimension. Fermat’s last theorem.

1900 1903 1907 1934

1900 1904 1921 1937

The chi-squared test The Poincaré conjecture Emmy Noether Alan Turing proposes his
is introduced by Karl is proposed, remaining publishes Ideal idea of a mathematical
Pearson, revolutionizing unproven for nearly Theory in Rings, a machine, which is
the field of statistics. a century. key text in the influential in the
development of rise of computers.
abstract algebra.

I
n 1900, as the arms race that the pseudonym Nicolas Bourbaki. number theory – explored the
led to World War I intensified, Starting from the basics, they met notion of multiple dimensions,
German mathematician David in the 1930s and 40s, rigorously and suggested spacetime as a
Hilbert attempted to anticipate the formalizing all branches of possible fourth dimension. Emmy
directions that mathematics would mathematics in terms of set theory. Noether, one of the first female
take in the 20th century. His list Others, notably Henri Poincaré, mathematicians of the modern era
of the 23 unsolved problems he explored the newly established field to gain recognition, came to the
considered crucial was influential of topology, the offshoot of geometry field of theoretical physics from a
in identifying the fields of dealing with surfaces and space. perspective of abstract algebra.
mathematics that could be fruitfully His famous conjecture concerns
explored by mathematicians. the 2-dimensional surface of a The computer age
3-dimensional sphere. Unlike many In the first half of the 20th century,
New century, new fields of his peers in the 20th century, applied mathematics was largely
One area of exploration was the Poincaré did not confine himself to concerned with theoretical physics,
foundations of mathematics. In any one single field of mathematics. especially the implications of
seeking to establish the logical As well as pure mathematics, he Einstein’s theories of relativity, but
basis of mathematics, Bertrand made significant discoveries in the latter part of the century was
Russell described a paradox that theoretical physics, including his increasingly dominated by advances
highlighted a contradiction in proposed principle of relativity. in computer sciences. Interest in
Georg Cantor’s naive set theory, Similarly, Hermann Minkowski – computing had begun in the 1930s,
leading to a reappraisal of the whose primary interest was in in the search for a solution to
subject. These ideas were taken geometry and the geometrical Hilbert’s Entscheidungsproblem
up by André Weil and others, using method applied to problems in (decision problem) and the
 MODERN MATHEMATICS 265

Edward Lorenz publishes Fermat’s last theorem

his work on chaos In the US, three is finally declared to be
theory that later mathematicians develop Benoit Mandelbrot solved after British
becomes synonymous the RSA algorithm, creates the Mandelbrot mathematician Andrew
with the example of using prime numbers to set, having coined the Wiles corrects an
the “butterfly effect”. encrypt information. term “fractal”. error in his initial proof.

1963 1977 1980 1995

1965 1977 1989 2006

Fuzzy logic is formulated The solution to the The World Wide Web Grigori Perelman’s
by Lotfi Zadeh and is soon four-colour problem invented by Tim proof of the Poincaré
used in a wide range of becomes the first Berners-Lee facilitates conjecture is fully
technologies, particularly mathematical theorem the rapid transmission of accepted by the
in Japan. to be proved by ideas, including mathematical
a computer. mathematics. community.

possibility of an algorithm to the principles more firmly with the was based on a binary system of
determine the truth or falsity of a aid of computer models. His visual logic first proposed by George
statement. One of the first to tackle images of attractors and oscillators, Boole in the 19th century, and the
the problem was Alan Turing, who along with Benoit Mandelbrot’s polar opposites of on-off, true-false,
went on to develop code-cracking fractals, became icons of these 0-1, and so on could not describe
machines during World War II that new fields of study. how things are in the real world.
were the forerunners of modern With the advent of computers, To overcome this, Lotfi Zadeh
computers. He later proposed a the secure transfer of data became suggested a system of “fuzzy”
test of artificial intelligence. an issue, and mathematicians logic, in which statements can
With the advent of electronic devised complex cryptosystems be partly true or false, in a range
computers, mathematics was in using the factorization of large between 0 (absolutely false) and 1
demand to provide methods of prime numbers. Launched in 1989, (absolutely true).
designing and programming the World Wide Web facilitated the In 2000, 21st-century
computer systems. But computers rapid transmission of knowledge, mathematics was heralded in
also provided a powerful tool for and computers became a part of a similar spirit to that of the
mathematicians. Hitherto unsolved everyday life, especially in the field 20th century, when the Clay
mathematical problems such as the of information technology. Mathematics Institute announced
four-colour theorem often involved seven Millennium Prize Problems,
lengthy calculations, which could New logic, new millennium offering a US$1 million prize for
now be done quickly and accurately For a while, it seemed electronic any of their solutions. As yet, only
by computer. Although Poincaré had computing could potentially the Poincaré conjecture has been
laid the foundations of chaos theory, provide answers to almost all solved; Grigori Perelman’s proof
Edward Lorenz was able to establish problems. But computing science was confirmed in 2006. ■
266

THE VEIL BEHIND

WHICH THE FUTURE
LIES HIDDEN
23 PROBLEMS FOR THE 20TH CENTURY

IN CONTEXT In 1900, David Hilbert stated 23 problems that he felt would

KEY FIGURE occupy mathematicians for the next century.
David Hilbert (1862–1943)
FIELDS
Logic, geometry
He felt solving these problems would improve
BEFORE our understanding of a wide range of fields,
1859 Bernhard Riemann including number theory, algebra,
proposes the Riemann geometry, and calculus.
hypothesis, a famous problem
that will later be Number 8 on
Hilbert’s list and remains
unresolved today.
10 of the Seven have
1878 Georg Cantor advances Two are too
problems solutions not Four remain
the continuum hypothesis, vague to ever
have been universally unresolved.
be resolved.
later Number 1 on Hilbert’s list. resolved. accepted.
AFTER

I
2000 The Clay Institute issues
a list of seven Millennium t requires a special technical thoughts in the decades to come.
brilliance and self-confidence This proved prescient; the maths
Prize mathematical problems,
to predict relevant problems for world rose to the challenge.
offering a million dollars for
the next hundred years, but this
each problem solved.
is what German mathematician The range of problems
2008 In a bid to stimulate David Hilbert did in 1900. Hilbert Many of Hilbert’s questions are
major new mathematical possessed a substantial grasp highly technical, but some are
breakthroughs, the US Defense of most fields of mathematics. At more accessible. Number 3, for
Advanced Research Projects the International Mathematical instance, asks if one of any two
Agency (DARPA) announces Congress in Paris in 1900, he polyhedra of the same volume can
its list of 23 unsolved problems. confidently announced his choice always be cut into finitely many
of 23 questions that he believed bits that can be reassembled
should occupy mathematicians’ to create the other polyhedron.
 MODERN MATHEMATICS 267
See also: Diophantine equations 80–81 ■ Euler’s number 186–91 ■ The Goldbach
conjecture 196 ■ The Riemann hypothesis 250–51 ■ Transfinite numbers 252–53

considered resolved, seven have

been partially solved, two have been
classed as too vague to ever be
definitively solved, three remain
unsolved, and one (also unsolved) is
The infinite! No other really a physics problem. Among the
question has ever moved unsolved problems is the Riemann
so profoundly the hypothesis, which some observers
spirit of man. think will remain unsolved for the
David Hilbert foreseeable future.
David Hilbert
Challenges for the future
Hilbert’s remarkable achievement Born in Prussia in 1862 to
was to accurately predict what German parents, David Hilbert
would concern mathematicians entered the University of
in the 20th century and beyond. Königsberg in 1880 and later
This was soon resolved in 1900 When American mathematician taught there before becoming
by German-born American and Fields Medal winner Steve professor of mathematics at
mathematician Max Dehn, who Smale came up with his own list the University of Göttingen in
concluded that it could not. of 18 questions in 1998, it included 1895. In this role, he turned
The continuum hypothesis, Hilbert’s eighth and 16th problems. Göttingen into one of the
the first problem on Hilbert’s list, Two years later, the Riemann mathematical hubs of the
world and taught a number of
pointed out that the set of natural hypothesis was also one of the
young mathematicians who
numbers (the positive integers) Clay Institute’s Millennium Prize later made their own mark.
was infinite, but so was the set problems. Today’s mathematicians Hilbert was renowned for
of real numbers between 0 and 1. face further challenges, but aspects his broad understanding of
As a result of the work of German of Hilbert’s problems – especially many areas of mathematics,
mathematician Georg Cantor, those that are still unsolved – and had a keen interest in
it was agreed that the first infinity remain relevant. ■ mathematical physics, too.
was “smaller” than the second. Exhausted by anaemia, he
The continuum hypothesis also retired in 1930, and Göttingen’s
stated that there was no infinity maths faculty soon declined
lying between these two infinities. after the Nazi purges of Jewish
Cantor himself was sure this was colleagues. Despite his great
true, but he could not prove it. In contribution to mathematics,
1940, Austrian–American logician Problem solving and theory Hilbert’s death in 1943,
building go hand in hand. during World War II, went
Kurt Gödel showed it could not
That’s why Hilbert risked largely unnoticed.
be proved that such an infinity
exists, and, in 1963, American offering a list of unsolved
mathematician Paul Cohen showed problems instead of presenting Key works
it could not be proved that such new methods or results.
an infinity does not exist. Hilbert’s Rüdiger Thiele 1897 Commentary on Numbers
1900 “The Problems of
first problem is substantially German mathematician
Mathematics” (Paris lecture)
resolved, although set theory (the
1932–35 Collected Works
study of the properties of sets) is a 1934–39 Foundations of
complex subject, and much more Mathematics (with Paul
work on it remains to be done. Bernays)
Of Hilbert’s 23 problems, 10 are
268
IN CONTEXT

STATISTICS
KEY FIGURE
Francis Galton (1822–1911)
FIELD

IS THE
Number theory
BEFORE
1774 Pierre-Simon Laplace
shows the expected pattern of

GRAMMAR
distribution around the norm.
1809 Carl Friedrich Gauss
develops the least squares
method of finding the best

OF SCIENCE
fit line for a scatter of data.
1835 Adolphe Quetelet
advocates the use of the bell
curve to model social data.
AFTER
THE BIRTH OF MODERN STATISTICS 1900 Karl Pearson proposes
the chi-squared test to
determine the significance of
differences between expected
and observed frequencies.

S
tatistics is the branch
of mathematics that is
concerned with analysing
and interpreting large quantities
of data. Its foundations were laid in
the late 19th century, principally by
British polymaths Francis Galton
and Karl Pearson.
Statistics investigates whether
the pattern of recorded data is
significant or random. Its origins
lie in the efforts of 18th-century
mathematicians such as Pierre-
Simon Laplace to identify
observational errors in astronomy.
In any set of scientific data, most
errors are likely to be very small,
and only a few are likely to be
very large. So when observations
are plotted on a graph, they create
a bell-shaped curve with a peak
created by the most likely result,
 MODERN MATHEMATICS 269
See also: Negative numbers 76–79 ■ Probability 162–65 ■ Normal distribution 192–93 ■ The fundamental theorem of
algebra 204–09 ■ Laplace’s demon 218–19 ■ The Poisson distribution 220–21

or “norm”, in the middle. In 1835,

Belgian mathematician Adolphe The quincunx
Quetelet posited that characteristics,
such as body mass, within a human
population follow a bell curve
pattern, in which values around
the mean are most frequent.
Higher and lower values are less
frequent. He devised the Quetelet
The probability of a
Index (now called the BMI) to ball falling on each
indicate body mass. side of a pin is ½.
1/1
Typically, plotting two variables,
Each ball is
such as height and age, on a graph dropped directly
1/2 1/2
creates a messy scatter of data The probability of a above the first pin.
points that cannot be linked by a ball falling either
neat line. However, in 1809, side of the middle
pin in the third row 1/4 2/4 1/4
German mathematician Carl is three times higher The probability
Friedrich Gauss found an equation than the far side of of a ball falling
to create a “best fit” line, which the outer pins. outside every
1/8 3/8 3/8 1/8 pin halves with
would show the relationship each row of pins.
between the variables. Gauss used
a method called “least squares”, 1/16 4 /16 6/16 4 /16 1/16
The probability
which involves adding up the of a ball falling
squares of the data; this is still into the middle
used by statisticians. By the 1840s, column is 1/32 5/32 10/32 10/32 5/32 1/32
highest, so The
mathematicians such as Auguste the biggest distribution
Bravais were looking at the level proportion of the balls
of error that could be accepted for of balls are 1/64 6/64 15/64 20/64 15/64 6/64 1/64 forms a
this line, and tried to pin down found here. bell curve.
the significance of the midpoint or
“median” of a set of data.

Correlation and regression

It was first Galton, then Pearson,
who began to draw these threads
together. Galton was inspired by
his cousin Charles Darwin’s work
on evolution, and his aim was to
show how likely it was that factors
such as height, physiognomy, and
even intelligence and criminal
tendencies might be passed from
one generation to the next. Galton
and Pearson’s ideas are tainted by
the doctrine of eugenics and racial
improvement, but the techniques Francis Galton invented the quincunx (sometimes
that they developed have found called the Galton board) to model the bell curve. His
applications elsewhere. ❯❯ original design had beads dropping over pegs.
270 THE BIRTH OF MODERN STATISTICS
Galton built an “anthropometric vary together. Regression,
laboratory” to collect information on on the other hand, looks for the
human characteristics, including head best equation for the graph line
size and quality of vision. It generated
huge amounts of data that he had to
for two variables, so that changes
analyse statistically. in one variable can be predicted
from changes to the other.

between them. His approach Standard deviation

involved establishing two related Although Galton’s main interest
concepts that are now at the heart was human heredity, he created
of statistical analysis: correlation a broad range of data sets.
and regression. Famously, he measured the size
Correlation measures the of seeds produced by sweet pea
degree to which two random plants grown from seven sets
variables, such as height and of seeds. Galton found that the
Galton was a rigorous scientist, weight, correspond. It often looks smallest pea seeds had larger
determined to analyse data to for a linear relationship – that is, offspring and the largest seeds
show mathematically how probable a relationship that gives a simple produced smaller offspring. He
outcomes are. In his innovative line on a graph, with one variable had discovered the phenomenon
1888 book Natural Inheritance, changing in step with the other. of “regression to the mean”, a
Galton showed how two sets of Correlation does not imply a causal tendency for measurements to
data can be compared to show if relationship between the two even out, always drifting towards
there is a significant relationship variables; it simply means they the mean over time.
Inspired by Galton’s work,
Pearson set out to develop the
Regression to the mean mathematical framework for
correlation and regression. After
1.8 m
exhaustive tests that involved
1.5 m tossing coins and drawing lottery
tickets, Pearson came up with
1.2 m
Parents the key idea of “standard deviation”,
0.9 m which shows how much on average
0.6 m observed values differ from
expected. To arrive at this figure, he
0.3 m found the mean, which is the sum of
0m

1.8 m

1.5 m

1.2 m
Children No observational problem
0.9 m
will not be solved by
0.6 m more data.
0.3 m
Vera Rubin
American astronomer
0m

Galton noticed that very tall parents tend to have children who are shorter
than their parents, while very short parents tend to have children who are
slightly taller than their parents. The second generation will be closer in
height than the first, an example of regression to the mean.
 MODERN MATHEMATICS 271
all the values divided by how
many values there are. Pearson Francis Galton introduced… Karl Pearson introduced…
then found the variance – the
average of the squared differences
from the mean. The differences
are squared in order to avoid
problems with negative numbers, The chi-
Standard squared
and the standard deviation is Correlation: Regression
deviation: test: for
the square root of the variance. the degree to to the mean:
the degree to variations
Pearson realized that by uniting which two the tendency
which results between
the mean and the standard variables of data to even
differ from observed and
deviation, he could calculate correspond. out over time.
the mean. expected
Galton’s regression precisely. data.

Chi-squared test
In 1900, after an extensive study
of betting data from the gaming Modern statistics was born.
tables of Monte Carlo, Pearson
described the chi-squared test,
now one of the cornerstones of
statistics. Pearson’s aim was to painstakingly worked out his table The combination of Galton’s
determine whether the difference by hand, but chi-squared tables correlation and regression, and
between observed values and are now produced using computer Pearson’s standard deviation and
expected values is significant, software. For each set of data, a chi-squared test, formed the
or simply the result of chance. chi-squared value can be found foundations of modern statistics.
Using his data on gambling, from the sum of all the differences These ideas have since been
Pearson calculated a table of between observed and expected refined and developed, but they
probability values, called chi- values. The chi-squared values are remain at the heart of data analysis.
squared (2), in which 0 shows checked against the table to find This is crucial in many aspects of
no significant difference from the significance of the variations modern life, from comprehending
expected (the “null hypothesis”), in the data within limits set by the economic behaviour to planning
whereas larger values show a researcher and known as “degrees new transport links and improving
significant difference. Pearson of freedom”. public health services. ■

Karl Pearson Karl Pearson was born in London at University College, London,
in 1857. An atheist, freethinker, in 1911. His views often led him
and socialist, he became one of into disputes. He died in 1936.
the greatest statisticians of the
20th century, but he was also a Key works
champion of the discredited
science of eugenics. 1892 The Grammar of Science
After graduating with a degree 1896 Mathematical
in mathematics from Cambridge Contributions to the Theory
University, Pearson became a of Evolution
teacher before making his mark in 1900 On the criterion that a
statistics. In 1901, he founded the given system of deviation from
statistical journal Biometrika with the probable in the case of a
Francis Galton and evolutionary correlated system of variables is
biologist Walter F. R. Weldon, such that it can be reasonably
followed by the world’s first supposed to have arisen from
university department of statistics random sampling.
272

A FREER LOGIC
EMANCIPATES US
THE LOGIC OF MATHEMATICS

IN CONTEXT The barber paradox imagines a town in which

all men must be clean shaven.
KEY FIGURE
Bertrand Russell
(1872–1970)
FIELD Any man who does not shave himself must be shaved
Logic by the town barber.
BEFORE
c. 300 bce Euclid’s Elements
contains an axiomatic
approach to geometry. So who shaves the barber?
1820s French mathematician
Augustin Cauchy clarifies the
rules for calculus, inaugurating
If he does not shave himself,
a new rigour in mathematics. If he shaves himself, he he is in the category of men
is not in the category of men who must be shaved by
AFTER who must be shaved by the the barber – another
1936 Alan Turing studies the barber – a contradiction. contradiction.
computability of mathematical
functions, with a view to

T
analysing which problems in
he common perception that Peano, and, in 1899, David Hilbert’s
mathematics can be decided
mathematics is logical, Foundations of Geometry. However,
and which cannot.
with fixed rules, evolved in 1903, Bertrand Russell published
1975 American logician over millennia, dating back to The Principles of Mathematics,
Harvey Friedman develops ancient Greece with the works which revealed a flaw in the logic
the “reverse mathematics” of Plato, Aristotle, and Euclid. A of one area of mathematics. In the
programme, which starts rigorous definition of the laws of book, he explored a paradox, known
with theorems and works arithmetic and geometry had as Russell’s paradox (or the Russell–
backwards to axioms. emerged by the 19th century, with Zermelo paradox, after German
the work of George Boole, Gottlob mathematician Ernst Zermelo, who
Frege, Georg Cantor, Giuseppe made a similar discovery in 1899).
 MODERN MATHEMATICS 273
See also: The Platonic solids 48–49 ■ Syllogistic logic 50–51 ■ Euclid’s Elements 52–57 ■ The Goldbach conjecture 196
■ The Turing machine 284–89

Bertrand Russell The son of a lord, Bertrand Russell rendered him morally unfit. He
was born in Monmouthshire, was awarded the Nobel Prize in
Wales, in 1872. He studied Literature in 1950, and in 1955
mathematics and philosophy at he and Albert Einstein released
Cambridge University, but was a joint manifesto calling for a
dismissed from an academic post ban on nuclear weapons. He
there in 1916 for anti-war later opposed the Vietnam War.
activities. A prominent pacifist Russell died in 1970.
and social critic, in 1918 he was
jailed for six months, during Key works
which he wrote his Introduction
to Mathematical Philosophy. 1903 The Principles of
Russell taught in the US in the Mathematics
1930s, although his appointment 1908 Mathematical Logic as
at a college in New York was Based on the Theory of Types
revoked due to a judicial 1910–13 Principia Mathematica
declaration that his opinions (with Alfred North Whitehead)

The paradox implied that set which placed restrictions on the Hilbert, Frege, and Peano
theory, which deals with the established model of set theory to develop complete logical
properties of sets of numbers or (known as “naive set theory”) by frameworks for mathematics
functions, and was fast becoming creating a hierarchy so that “the were destined to have logical
the bedrock of mathematics, set of all sets” would be treated gaps, however watertight they
contained a contradiction. To differently from its constituent tried to make them.
explain the problem, Russell used smaller sets. In so doing, Russell Gödel’s theorem also implied
an analogy known as the barber managed to circumvent the that some as-yet unproven
paradox in which a barber shaves paradox completely. He utilized theorems in mathematics, such
every man in town apart from those this new set of logical principles as the Goldbach conjecture, may
who shave themselves, creating in the momentous Principia never be proved. This has not,
two sets of people: those who Mathematica, written with Alfred however, deterred mathematicians
shave themselves and those shaved North Whitehead and published in in their resolute efforts to prove
by the barber. However, this begs three volumes from 1910 to 1913. Gödel wrong. ■
the question: if the barber shaves
himself, to which of the two sets Logical gaps
does the barber belong? In 1931, Kurt Gödel, an Austrian
Russell’s barber paradox mathematician and philosopher,
contradicted Frege’s Basic Laws published his incompleteness
of Arithmetic concerning the logic theorem (following on from his
of mathematics, which Russell had completeness theorem of a few Every good mathematician is
pointed out in a letter to Frege in years earlier). The 1931 theorem at least half a philosopher, and
1902. Frege declared that he was concluded that there will always every good philosopher is at
“thunderstruck”, and he never found exist some statements regarding least half a mathematician.
an adequate solution to the paradox. numbers that may be true but can Gottlob Frege
never be proved. Furthermore,
A theory of types expanding mathematics by simply
Russell went on to produce adding more axioms will lead to
his own response to his paradox, further “incompleteness”. This
developing a “theory of types”, meant that the efforts of Russell,
274

THE UNIVERSE IS
FOUR-DIMENSIONAL
MINKOWSKI SPACE

T
here are three dimensions It was in the 18th century that
IN CONTEXT in our familiar view of the scientists first began questioning
world – length, width, and whether three-dimensional
KEY FIGURE
height – and they can largely be Euclidean geometry could
Hermann Minkowski
described mathematically by the describe the entire Universe.
(1864–1909) geometry of Euclid. But in 1907, Mathematicians started to
FIELD German mathematician Hermann develop non-Euclidean geometric
Geometry Minkowski delivered a lecture in frameworks, while some considered
which he added time, an invisible time as a potential dimension.
BEFORE fourth dimension, to create the Light provided the mathematical
c. 300 bce In his book concept of spacetime. This has prompt. In the 1860s, Scottish
Elements, Euclid establishes played a key part in understanding
the geometry of 3-D space. the nature of the Universe. It has
A black hole occurs when spacetime
provided a mathematical framework warps so much that its curvature
1904 In his book The for Einstein's theory of relativity,
Fourth Dimension, British becomes infinite at the hole’s centre.
allowing scientists to develop and Even light is not fast enough to escape
mathematician Charles Hinton expand this theory. the hole’s immense gravitational pull.
coins the term “tesseract” for
a four-dimensional cube.
1905 French scientist Henri
Poincaré has the idea of
making time the fourth
dimension in space.
1905 Albert Einstein states
his theory of special relativity.
AFTER
1916 Einstein writes the key
paper outlining his theory of
general relativity, in which he
explains gravity as a curvature
of spacetime.
 MODERN MATHEMATICS 275
See also: Euclid’s Elements 52–57 ■ Newton’s laws of motion 182–83 ■ Laplace’s
demon 218–19 ■ Topology 256–59 ■ Proving the Poincaré conjecture 324–25

Stationary object Moving objects Object moving at

the speed of light

Time Time Faster moving Time

object
Slower
moving
object As nothing
travels faster 45˚
than light, the
Space Space worldline of all Space
objects is in
this section.
Hermann Minkowski
The worldline for a A slower object has a This worldline has a
stationary object is steeper worldline as it is 45º angle, with a 1:1 Born in Aleksotas (now in
vertical because it is not progressing more slowly ratio between the
Lithuania) in 1864, Minkowski
moving through space. along the space axis. time and space axes.
moved with his family to
Königsberg in Prussia in 1872.
scientist James Clerk Maxwell “worldline”, which could be plotted As a boy, he showed an
found that the speed of light is the on a graph, with space and time as aptitude for maths and began
same whatever the speed of its the axes. A static object produces a his studies at the University
source. Mathematicians then vertical worldline, and the worldline of Königsberg aged 15. By 19,
developed his equations to try to of a moving object is at an angle (see he had won the Paris Grand
understand how the finite speed above). The worldline angle of an Prix for mathematics, and at
23, he became a professor
of light fitted into the coordinate object moving at the speed of light
at the University of Bonn. In
system of space and time. is 45°. According to Minkowski, no
1897 he taught the young
worldline can exceed this angle, but Albert Einstein in Zurich.
Mathematics of relativity in reality, there are three axes of Following a move to
In 1904, Dutch mathematician space, plus the axis of time, so the Göttingen in 1902, Minkowski
Henrik Lorentz developed a set of 45° worldline is really a “hypercone”, became fascinated by the
equations, called “transformations”, a 4-dimensional figure. All physical mathematics of physics,
to show how mass, length, and reality is held within it, as nothing especially the interaction
time change as a spatial object can travel faster than light. ■ of light and matter. When
approaches the speed of light. A Einstein unveiled his theory
year later, Albert Einstein produced of special relativity in 1905,
his theory of special relativity, Minkowski was spurred on
which proved that the speed of to develop his own theory,
light is the same throughout the in which space and time form
Universe. Time is a relative, not an part of a four-dimensional
Henceforth, space by itself, reality. This concept inspired
absolute, quantity – running at
and time by itself shall fade Einstein’s theory of general
different speeds in different places relativity in 1915, but by then,
and woven together with space. to mere shadows, and only Minkowski was dead – killed
Minkowski turned Einstein’s some union of the two will at 44 years old by a ruptured
theory into mathematics. He preserve independent reality. appendix.
showed how space and time Hermann Minkowski
are parts of a four-dimensional
Key work
spacetime, where each point in
space and time has a position. He 1907 Raum und Zeit (Space
represented movement between and Time)
positions as a theoretical line, a
276

RATHER A
DULL NUMBER
TAXICAB NUMBERS

A
“taxicab” number, Ta(n), is of this story ensured that 1,729
IN CONTEXT the smallest number that would become one of the best-
can be expressed as the known numbers in mathematics.
KEY FIGURE
sum of two positive cubed integers Ramanujan was not the first to
Srinivasa Ramanujan
(whole numbers) in n (number of) make note of this number’s unique
(1887–1920) different ways. They owe their name properties; French mathematician
FIELD to an anecdote from 1919, when Bernard Frénicle de Bessy had
Number theory British mathematician G. H. Hardy also written about them in the
went to Putney, London, to visit his 17th century.
BEFORE protégé Srinivasa Ramanujan, who
1657 In France, mathematician was ill. Arriving in a cab with the Extending the concept
Bernard Frénicle de Bessy number 1,729, Hardy remarked, The taxicab story inspired later
cites the properties of 1,729, “Rather a dull number, don’t you mathematicians to examine the
the original “taxicab” number. think?” Ramanujan disagreed, then property that Ramanujan had
explained that 1,729 is the smallest recognized and to expand its
1700s Swiss mathematician number that is the sum of two application. The hunt was on for
Leonhard Euler calculates positive cubes in two different the smallest number that could
that 635,318,657 is the ways. Hardy’s frequent retelling be expressed as the sum of two
smallest number that can be
expressed as the sum of two
fourth powers (numbers to
the power of 4) in two ways. 1,729 is the smallest number that is the sum of two positive
cubes in two different ways.
AFTER
1978 Belgian mathematician
Pierre Deligne receives the
Fields Medal for his work on
number theory, including the 13 + 123 = 1,729
103 + 93 = 1,729
proof of a conjecture in the
theory of modular forms that
was first made by Ramanujan.

It is not a dull number.

 MODERN MATHEMATICS 277
See also: Cubic equations 102–05 ■ Elliptic functions 226–27 ■ Catalan’s
conjecture 236–37 ■ The prime number theorem 260–61

Does Ta(n) always exist?

The existence of Ta(n) was proved theoretically in 1938 for all
values of n, but the search is still on for larger taxicab numbers.
Even with the benefits of computer calculations, mathematicians
have not yet moved beyond Uwe Hollerbach’s discovery of Ta(6).

Date Number Value Discoverer

N/A Ta(1) 2 N/A
1657 Ta(2) 1,729 de Bessy
Srinivasa Ramanujan
1957 Ta(3) 87,539,319 Leech
Born in Madras, India in 1887,
Rosenstiel, Ramanujan displayed an
1989 Ta(4) 6,963,472,309,248 Dardis, extraordinary aptitude for
Rosenstiel mathematics at an early age.
1994 Ta(5) 48,988,659,276,962,496 Dardis Finding it hard to get full
recognition locally, he took the
2008 Ta(6) 24,153,319,581,254,312,065,344 Hollerbach bold step of sending some of
his results to G. H. Hardy, then
a professor at Trinity College,
positive cubes in three, four, or represented the number of prime Cambridge. Hardy declared
that they had to be the work
more different ways. A further numbers less than x; Ramanujan
of a mathematician “of the
question was whether Ta(n) exists was unable, however, to offer a
highest class”, and had to be
for all values of n; in 1938, Hardy rigorous proof. true, because no one could
and British mathematician Edward Taxicab numbers have little invent them. In 1913, Hardy
Wright proved that it does (an practical use, but they still invited Ramanujan to work
existence proof), but developing inspire scholars as curiosities. with him in Cambridge. The
a method of finding Ta(n) in each Mathematicians now also seek collaboration was hugely
case has proved elusive. “cabtaxi” numbers: based on the productive: in addition to the
Extending the concept further, taxicab formula, these allow taxicab numbers, Ramanujan
the expression Ta(j, k, n) seeks the calculations using both positive also developed a formula for
smallest positive number that is and negative cubes. ■ obtaining the value of pi to
the sum of any number of different a high level of accuracy.
positive integers (j), each to any However, Ramanujan
power (k) in n distinct ways. For suffered from ill health. He
example, Ta(4, 2, 2) requires the returned to India in 1919 and
died a year later – probably as
smallest number that is the sum
a result of amoebic dysentery
of four squares (or two fourth powers) contracted years earlier. He
in two different ways: 635,318,657. An equation means
left behind several notebooks,
nothing to me unless it which mathematicians are
Continuing relevance expresses a thought of God. still studying today.
Taxicab numbers were only one Srinivasa Ramanujan
area of Hardy and Ramanujan’s
Key work
work. Their main focus was prime
numbers. Hardy was excited by 1927 Collected papers of
Ramanujan’s claim that he had Srinivasa Ramanujan
found a function of x that exactly
278

A MILLION
MONKEYS BANGING
ON A MILLION
TYPEWRITERS
THE INFINITE MONKEY THEOREM

I
n the early 20th century, Earth was created by atoms
IN CONTEXT French mathematician Émile coming together entirely by
Borel explored improbability – chance. Three centuries later, the
KEY FIGURE
when events had a very small Roman philosopher Cicero argued
Émile Borel (1871–1956)
chance of ever occurring. Borel that this was so unlikely that it
FIELD concluded that events with a was essentially impossible.
Probability sufficiently small probability will
never occur. He was not the first to Defining impossibility
BEFORE study the probability of unlikely Over the past two millennia,
45 bce The Roman philosopher events. In the 4th century bce, the various thinkers have probed the
Cicero argues that a random ancient Greek philosopher Aristotle balance between the improbable
combination of atoms forming suggested in Metaphysics that and the impossible. In the 1760s,
Earth is highly improbable.
1843 Antoine Augustin
Cournot makes a distinction In an infinite amount of time, an infinite number
between physical and of events will happen.
practical certainty.
AFTER
1928 British physicist Arthur
Eddington develops the idea
that improbable is impossible. A monkey typing for infinity
The monkey would therefore
would produce every letter in
2003 Scientists at Plymouth produce every finite text an
every possible combination
University in the UK test Borel’s infinite number of times.
an infinite number of times.
theory with real monkeys and
a computer keyboard.
2011 American programmer
Jesse Anderson’s million
virtual monkey software According to mathematical probability, a
generates the complete monkey typing for infinity will eventually type
works of Shakespeare. the complete works of Shakespeare.
 MODERN MATHEMATICS 279
See also: Probability 162–65 ■ The law of large numbers 184–85 ■ Normal distribution 192–93 ■ Laplace’s demon 218–19
■ Transfinite numbers 252–53

its tip. He argued that it is possible

but highly unlikely, and made the
distinction between a physical
certainty – an event that can
happen physically, like the
The physically impossible balancing cone – and a practical
event is therefore the one that certainty, which is so unlikely that
has infinitely small probability, in practical terms it is considered
and only this remark gives impossible. In what is sometimes
substance… to the theory of known as Cournot’s principle,
mathematical probability. Cournot suggested that an event
Antoine Augustin Cournot with a very small probability will
not happen.
Borel’s theory is often applied to
stock markets, where the level of chaos
Infinite monkeys means that in some cases random
Borel’s law, which he called the law selection performs better than selection
of single chance, gave a scale to based on traditional economic theories.
practical certainty. For events on
French mathematician Jean a human scale, Borel considered
d’Alembert questioned whether it events with a probability of less than noted that, while it cannot be
was possible to have a very long 10-6 (or 0.000001) to be impossible. mathematically proven that it is
string of occurrences in a sequence He also came up with a famous impossible for monkeys to type
in which occurrence and non- example to illustrate impossibility: Shakespeare, it is so unlikely
occurrence are equally likely – for monkeys hitting typewriter keys that mathematicians should
example, whether a person flipping at random will eventually type the consider it impossible. This idea
a coin might get “heads” two complete works of Shakespeare. of monkeys typing the works of
million times in a row. In 1843, This outcome is highly improbable, Shakespeare captured people’s
French mathematician Antoine but mathematically, over an infinite imagination and Borel’s law
Augustin Cournot questioned the time (or with an infinite number came to be known as the infinite
possibility of balancing a cone on of monkeys), it must happen. Borel monkey theorem. ■

Émile Borel Born in 1871 in Saint-Affrique, particular output. During World

France, Émile Borel was a War I, Borel worked for the War
mathematics prodigy and Office and later became minister
graduated top of his class from the of the navy. Imprisoned when
École Normale Supérieure in 1893. the Germans invaded France in
After lecturing in Lille for four World War II, he was released
years, he returned to the École, and fought for the Resistance,
where he dazzled fellow earning himself the Croix de
mathematicians with a series Guerre. He died in 1956 in Paris.
of brilliant papers.
Borel is best known for his Key works
infinite monkey theorem, but his
lasting achievement was in laying 1913 Le Hasard (Chance)
the foundations for the modern 1914 Principes et formules
understanding of complex classiques du calcul des
functions – what a variable probabilités (Principles and
must be altered by to achieve a classic formulas of probability)
280

SHE CHANGED
THE FACE OF
ALGEBRA
EMMY NOETHER AND ABSTRACT ALGEBRA

I
n the 19th century, analysis mathematicians such as Joseph-
IN CONTEXT and geometry were the leading Louis Lagrange, Carl Friedrich
fields of mathematics, while Gauss, and British mathematician
KEY FIGURE
algebra was considerably less Arthur Cayley, but gained traction
Emmy Noether (1882–1935)
popular. Throughout the Industrial when German mathematician
FIELD Revolution, applied mathematics Richard Dedekind began to
Algebra was prioritized over areas of study study algebraic structures. He
that were more theoretical. This all conceptualized the ring – a set of
BEFORE changed in the early 20th century elements with two operations, such
1843 German mathematician with the rise of “abstract” algebra, as addition and multiplication. A
Ernst Kummer develops the which became one of the key fields ring can be broken into parts called
concept of ideal numbers – of mathematics, largely thanks “ideals” – a subset of elements. For
ideals in the ring of integers. to the innovations of German example, the set of odd integers are
mathematician Emmy Noether. an ideal in the ring of integers.
1871 Richard Dedekind builds
Noether was not the first to focus
on Kummer’s idea to formulate on abstract algebra. Work on algebra Significant works
definitions of rings and ideals theory had been developed by Noether began her work on abstract
more generally. algebra shortly before World War I
1890 David Hilbert refines with her exploration of invariant
the concept of the ring. theory, which explained how
some algebraic expressions stay the
AFTER same while other quantities change.
1930 Dutch mathematician In 1915, this work led her to make
Bartel Leendert Van der My methods are really a major contribution to physics;
Waerden writes the first methods of working she proved that the laws of
comprehensive treatment and thinking; this is conservation of energy and mass
of abstract algebra. why they have crept in each correspond to a different type
everywhere anonymously. of symmetry. The conservation of
1958 British mathematician Emmy Noether electric charge, for example, is
Alfred Goldie proves that related to rotational symmetry.
Noetherian rings can be Now called Noether’s theorem, it
understood and analysed in was praised by Einstein for the
terms of simpler ring types. way it addressed his theory of
general relativity.
 MODERN MATHEMATICS 281
See also: Algebra 92–99 ■ The binomial theorem 100–01 ■ The algebraic resolution of equations 200–01
■ The fundamental theorem of algebra 204–09 ■ Group theory 230–33 ■ Matrices 238–41 ■ Topology 256–59

Mathematicians created a system called abstract

algebra to generalize mathematical objects and the operations
that act on them.

A set is a collection of objects or elements, such as integers.

Emmy Noether
A group is a type of set that includes an Born in 1882, Emmy Noether
operation (e.g. addition), and follows certain axioms. struggled to find education,
recognition, and even basic
employment in early 20th
century academia as a Jewish
woman in Germany. Although
her mathematical skill won her
a position at the University of
A ring is a type of group that includes a second Erlangen – where her father
operation, often multiplication. It also includes the axiom of also taught mathematics –
associativity, whereby each of the operations can be applied in from 1908 to 1923 she received
any order without affecting the result. no pay. She later faced similar
discrimination in Göttingen,
where her colleagues had to
fight to have her officially
included in the faculty. In
1933, the rise of the Nazis
led to her dismissal, and she
Noether’s contributions to moved to the US, working at
ring theory furthered our understanding Bryn Mawr College and at the
of algebraic structures. Institute for Advanced Study
until her death in 1935.

Key works
In the early 1920s, Noether’s work result. In a 1924 paper, she proved
focused on rings and ideals. In that in these commutative rings, 1921 Idealtheorie in
a key paper in 1921, Idealtheorie every ideal is the unique product Ringbereichen (Ideal Theory
in Ringbereichen (Ideal Theory in of prime ideals. One of the most in Rings)
1924 Abstrakter Aufbau der
Rings), she studied ideals in a brilliant mathematicians of her
Idealtheorie im algebraischen
particular set of “commutative time, Noether laid the foundations Zahlkörper (Abstract
rings”, in which the numbers can for the development of the entire Construction of Ideal Theory
be swapped around when they are field of abstract algebra with her in Algebraic Fields)
multiplied without affecting the contributions to ring theory. ■
282

STRUCTURES ARE
THE WEAPONS OF THE
MATHEMATICIAN
THE BOURBAKI GROUP

R
ussian mathematical libraries and countless students
IN CONTEXT genius Nicolas Bourbaki of mathematics have learned the
was one of the most prolific tools of their trade from his work.
KEY FIGURES
and influential mathematicians of Bourbaki, however, never
André Weil (1906–1998),
the 20th century. His monumental existed. He was a fiction created
Henri Cartan (1904–2008) work Éléments de Mathématique in the 1930s by young French
FIELDS (Elements of Mathematics, 1960), mathematicians who were striving
Number theory, algebra occupies a key place in university to fill the vacuum left by the

BEFORE
1637 René Descartes creates
coordinate geometry, allowing A group of French mathematicians felt disheartened by the
points on a flat surface to state of French mathematics, and wanted to…
be described.
1874 Georg Cantor creates set
theory, describing how sets and
their subsets interrelate. …end reliance
…take a more …think about
1895 Henri Poincaré lays rigorous approach on creative algebra in terms of
the foundations of algebraic to mathematics. guesswork. geometric shapes.
topology in Analysis Situs
(Analysis of Position).
AFTER
1960s The New Mathematics However, they were afraid of retaliation, and wanted to
movement, which focuses operate in secret, so…
on set theory, becomes
popular in American and
European schools.
1995 Andrew Wiles publishes
his final proof of Fermat’s …they published their writings under the
last theorem. pseudonym of Nicolas Bourbaki.
 MODERN MATHEMATICS 283
See also: Coordinates 144–51 ■ Topology 256–59 ■ The butterfly effect 294–99 ■ Proving Fermat’s last theorem 320–23
■Proving the Poincaré conjecture 324–25

The Bourbaki group poses for a

photo at the first Bourbaki congress
in July 1935. Among them are Henri
Cartan (standing far left) and André
Weil (standing fourth from left).

The group aimed to strip

mathematics back to basics and
provide a foundation from which it
could go forward. While their work
sparked a brief fad in the 1960s,
it proved too radical for teachers
and pupils alike. The group was
often at odds with cutting-edge
mathematics and physics, and
was so focused on pure maths
that applied maths was of little
interest to them. Topics containing
uncertainty, such as probability,
devastation of World War I. While The group – which included Claude had no place in Bourbaki’s work.
other countries had kept academics Chevalley, Jean Delsarte, Jean Even so, the group made
at home, French mathematicians Dieudonné, and René de Possel – important contributions across
had joined their countrymen in agreed to create a new body of a wide range of mathematical
the trenches and a generation of work that covered all fields of topics, particularly in set theory
teachers had been killed. French mathematics. Meeting regularly and algebraic geometry. The group,
mathematics was stuck with and marshalled by Dieudonné, the which acts in secrecy and whose
antiquated textbooks and teachers. group produced book after book, members must resign at age 50,
led by Éléments de Mathématique. still exists, although Bourbaki now
Renewing mathematics Their work was likely to be publishes infrequently. The most
Some young teachers believed controversial, so they adopted the recent two volumes were published
that French mathematics had fallen pseudonym Nicolas Bourbaki. in 1998 and 2012. ■
victim to a lack of rigour and
precision. They were distrustful Bourbaki’s legacy
of the creative guesswork, as they
saw it, of older mathematicians Topology and set theory – the perhaps their lasting legacy.
such as Henri Poincaré in meeting between numbers and It was at least partly Bourbaki’s
developing chaos theory and shapes – were for Bourbaki at work on algebraic geometry
mathematics for physics. the very root of mathematics that led British mathematician
In 1934, two young lecturers at and lay at the heart of the Andrew Wiles to finally prove
the University of Strasbourg, André group’s work. René Descartes Fermat’s last theorem; he
Weil and Henri Cartan, took matters had first made the link between published his proof in 1995.
in hand. They invited six fellow shapes and numbers in the Some mathematicians
17th century with coordinate believe algebraic geometry
former students from the École
geometry, turning geometry into has great untapped potential
Normale Supérieur to lunch in algebra. Bourbaki helped make for the future. It already has
Paris, hoping to persuade them to the link the other way, turning real-world applications such as
take part in an ambitious project – algebra into geometry to create in programming codes in mobile
to write a new treatise that would algebraic geometry, which is phones and smart cards.
revolutionize mathematics.
A SINGLE MACHINE TO
COMPUTE
ANY COMPUTABLE
SEQUENCE
THE TURING MACHINE
286 THE TURING MACHINE

IN CONTEXT
KEY FIGURE Computing the answers to many number problems can be
reduced to an algorithm – a sequence of mathematical steps
Alan Turing (1912–54) that are applied in a predefined order.
FIELD
Computer science
BEFORE
1837 In the UK, Charles
Babbage designs the
Analytical Engine, a The Turing machine
Some algorithms
mechanical computer using can process any
reach answers;
the decimal system. If it had algorithm, solvable
others loop forever.
or not.
been constructed, it would
have been the first “Turing-
complete” device.
AFTER
1937 Claude Shannon designs
electrical switching circuits
that use Boolean algebra to By inputting algorithms to the
make digital circuits that machine, it is possible to prove when an
follow rules of logic. algorithm has no answer.
1971 American mathematician
Stephen Cook poses the P

A
versus NP problem, which lan Turing is often cited using a given set of instructions
tries to understand why some as the “father of digital in a given order – would arrive
mathematical problems can computing”, yet the at a solution to the problem.
quickly be verified but would Turing machine that earned him In 1931, Austrian mathematician
take billions of years to prove, that accolade was not a physical Kurt Gödel demonstrated that
despite computers’ immense device but a hypothetical one. mathematics based on formal
calculating power. Instead of constructing a prototype axioms could not prove everything
computer, Turing used a thought that was true according to those
experiment in order to solve the axioms. What Gödel called the
Entscheidungsproblem (decision “incompleteness theorem” found
problem) that had been posed by that there was a mismatch
German mathematician David between mathematical truth
Hilbert in 1928. Hilbert was and mathematical proof.
interested in whether logic could
be made more rigorous by being Ancient roots
If a machine is expected simplified into a set of rules, or Algorithms have ancient origins.
to be infallible, it cannot axioms, in the same way that One of the earliest examples is
also be intelligent. arithmetic, geometry, and other the method used by the Greek
Alan Turing fields of mathematics were thought geometer Euclid to calculate the
possible to simplify at the time. greatest common divisor of two
Hilbert wanted to know if there numbers – the largest number
was a way to predetermine whether that divides both of them without
an algorithm – a method for solving leaving a remainder. Another early
a specific mathematical problem example is Eratosthenes’ sieve,
 MODERN MATHEMATICS 287
See also: Euclid’s Elements 52–57 ■ Eratosthenes’ sieve 66–67 ■ 23 Problems for
the 20th century 266–67 ■ Information theory 291 ■ Cryptography 314–17

Numbers, with an Application to the

Entscheidungsproblem”. It showed
that there is no solution to Hilbert’s
decision problem: some algorithms
are not computable, but there is no
A man provided with universal mechanism for identifying
paper, pencil, and rubber, them before trying them.
and subject to strict Turing reached this conclusion
discipline, is in effect a using his hypothetical machine,
universal machine. which came in two parts. First
Alan Turing there was a tape, as long as it
Alan Turing
needed to be, divided into sections,
each section carrying a coded Born in London in 1912, Alan
character. This character could be Turing was described as a
anything, but the simplest version genius by his teachers. After
used 1s and 0s. The second part graduating with a first-class
was the machine itself, which read degree in mathematics from
attributed to the 3rd-century bce the data from each section of the the University of Cambridge
Greek mathematician. It is an tape (either by the head or tape in 1934, he went on to study
algorithm for sorting primes from moving). The machine would be at Princeton in the US.
composite (not prime) numbers. equipped with a set of instructions Returning to the UK in
The algorithms of Eratosthenes and (an algorithm) that controlled the 1938, Turing joined the
Government Code and Cypher
Euclid work perfectly and can be behaviour of the machine. The
School at Bletchley Park. After
proven always to do so, but they did machine (or tape) could move left, ❯❯ war broke out in 1939, he and
not conform to a formal definition. It others developed the Bombe,
was the need for this that led Turing an electromechanical device
Clerks at work in Hut 8, Bletchley
to create his “virtual machine”. Park, UK, during World War II. At one that deciphered enemy
In 1937, Turing published his first point, Turing led the work of Hut 8, messages. Following the war,
paper as a fellow of King’s College, which deciphered communiqués Turing worked at Manchester
Cambridge, “On Computable between Adolf Hitler and his forces. University, where he designed
the Automatic Computing
Engine (ACE) and developed
further digital devices.
In 1952, Turing was
convicted of homosexuality,
then a crime in the UK. He
was also barred from working
on codebreaking for the
government. To avoid prison,
Turing agreed to hormone
treatment to reduce his libido.
In 1954, he committed suicide.

Key work

1939 “Report on the

Applications of Probability
to Cryptography”
288 THE TURING MACHINE
The Turing machine consists of a head that reads data from an because it was impossible to
infinitely long tape. The machine’s algorithm might either instruct the know if the machine would ever
head or the tape to move – to go left, right, or stay still. The memory halt or not, then the answer to
keeps track of changes and feeds them back into the algorithm.
the decision problem was No:
there was no universal test for
the validity of algorithms.

1 0 1 0 1 0 0 1 0 1 0 1 Computer architecture
READ/WRITE The Turing machine had not
HEAD finished its job. Turing and others
Tape
realized that this simple concept
could be used as a “computer”.
At the time, the term “computer”
was used to describe a person who
carried out complex mathematical
calculations. A Turing machine
MEMORY ALGORITHM would do so using an algorithm to
rewrite an input (the data on the
tape) into an output. In terms of
computing ability, the algorithms
at work in a Turing machine are
right, or stay where it was, and it take any algorithm and test it using the strongest type ever devised.
could rewrite the data on the tape, the machine to see if it halted. Modern computers and the
switching a 0 to 1 or vice versa. In essence, the Turing machine programs that run on them are
Such a machine could carry out (M) is an algorithm that tests effectively working as Turing
any conceivable algorithm. another algorithm (A) to see if it machines, and so are said to be
Turing was interested in is solvable. It does this by asking: “Turing complete”.
whether any algorithm put into the does A halt (have a solution)? M As a leading figure in
machine would cause the machine then reaches an answer of Yes or mathematics and logic, Turing
to halt. Halting would signify that No. Turing then imagined a made important contributions to
the algorithm had arrived at a modified version of this machine the development of real computers,
solution. The question was whether (M*), which would be set up so that not just virtual ones. However, it
there was a way of knowing which if the answer was Yes (A does halt),
algorithms (or virtual machines), then M* would do the opposite – it
would halt and which would not; would loop forever (and not halt). If
if Turing could find out, he would the answer was No (A does not
answer the decision problem. halt), then M* would halt.
Turing then took this thought We need to feed [information]
The halting problem experiment further by imagining through a processor. A human
Turing approached this problem as that you could use the machine must turn information into
a thought experiment. He began by M* to test whether its own intelligence or knowledge.
imagining a machine that was able algorithm, M*, would halt. If the We’ve tended to forget that
to say whether any algorithm (A) answer was Yes, the algorithm M*
no computer will ever ask
would halt (provide an answer and will halt, then the machine M*
stop running) when given an input would not halt. If the answer was
a new question.
to which the answer was either Yes No, the algorithm M* never halts,
Grace Hopper
American computer scientist
or No. Turing was not concerned then the machine M* would halt.
with the physical mechanics of Turing’s thought experiment
such a machine. Once he had had, therefore, created a paradox
conceptualized such a machine, which could be used as a form of
however, he could theoretically mathematical proof. It proved that,
 MODERN MATHEMATICS 289
binary data. Initially referred to as
“discrete variables”, in 1948 the 1s
and 0s in computer code were
renamed “bits”, short for binary
digits. This term was coined by
Claude Shannon, a leading figure The popular view that
in information theory – the field scientists proceed inexorably
of mathematics examining how from well-established fact to
information could be stored and well-established fact, never
transmitted as digital codes. being influenced by any
Early computers used multiple unproved conjecture, is
bits as “addresses” for sections quite mistaken.
of memory – showing where the Alan Turing
processor should look for data.
These chunks of bits became
known as “bytes”, spelled this way
to avoid confusion with “bits”. In
the early decades of computing,
A Turing Bombe, used to decipher
coded messages, has been rebuilt bytes generally contained 4 or 6
at the museum at Bletchley Park, bits, but the 1970s saw the rise of processed by these algorithms
the British code-breaking centre Intel’s 8-bit microprocessors, and into outputs, such as text on a
during World War II. byte became the unit for 8 bits. The device’s screen.
8-bit byte was convenient because 8 The principles of the Turing
bits have 28 permutations (256), and machine are still used in modern
was Hungarian mathematician can encode numbers from 0 to 255. computers and look set to continue
John von Neumann who contrived Armed with a binary code until quantum computing changes
a real-life version of Turing’s arranged in sets of eight digits – how information is processed. A
hypothetical device using a and later even longer strings – classical computer bit is either 1
central processing unit (CPU) that software could be produced for any or 0, never anything in between.
converted an input to an output by conceivable application. Computer A quantum bit, or “qubit”, uses
calling up information stored in an programs are simply algorithms; superposition to be both a 1 and
internal memory and sending back the inputs from a keyboard, 0 at the same time, which boosts
new information to be saved. He microphone, or touchscreen are computing power enormously. ■
proposed his configuration, known
as the “von Neumann architecture”, The Turing test win the prize. The AIs must
in 1945, and today, a similar fool human judges into thinking
process is used in almost every In 1950, Turing developed a test they are human rather than a
computing device. of a machine’s ability to exhibit computer program. AIs who
intelligent behaviour equivalent progress to the final take it in
Binary code to, or indistinguishable from, turns to communicate with one
Turing did not initially envisage that of a human. In his view, if of four judges. Each judge is also
that his machine would use only a machine appeared to be communicating with a human
binary data. He merely thought it thinking for itself, then it was. and must decide whether the AI
would use code with a finite set of The annual Loebner Prize or the human is most humanlike.
characters. However, binary was in Artificial Intelligence (AI) Over the years the test has
was inaugurated in 1990 by had many critics, who question
the language of the first Turing-
American inventor Hugh its ability to truly judge the
complete machine ever built, the
Loebner and the Cambridge intelligence of an AI effectively
Z3. Constructed in 1941 by German Center for Behavioral Studies, or see the competition as a
engineer Konrad Zuse, the Z3 Massachusetts. Every year, stunt that does not advance
used electromechanical relays, or computers using AI try to knowledge in the field of AI.
switches, to represent 1s and 0s of
290

SMALL THINGS ARE

MORE NUMEROUS
THAN LARGE
BENFORD’S LAW
THINGS

I
t might be expected that law better than data that is more
IN CONTEXT in any large set of numbers, closely grouped. The numbers in the
those that start with the Fibonacci sequence follow Benford’s
KEY FIGURE
digit 3 would occur with roughly law, as do the powers of many
Frank Benford (1883–1948)
the same frequency as those that integers. Numbers that act as a
FIELD start with any other digit. However, name or label, such as bus or
Number theory many sets of numbers – a list of telephone numbers, do not fit.
populations for UK villages, towns, When numbers are made up,
BEFORE and cities, for example – show a they tend to have a more equal
1881 Canadian astronomer distinctly different pattern. Often distribution of leading digits than
Simon Newcomb notices that in a set of naturally occurring if they followed Benford’s law. This
the pages most often referred numbers, around 30 per cent of has enabled investigators to use
to in logarithm tables are for the numbers have a leading digit of the law to detect financial fraud. ■
numbers starting with 1. 1, around 17 per cent have a leading
digit of 2, and less than 5 per cent
AFTER have a leading digit of 9. In 1938,
1972 Hal Varian, an American American physicist Frank Benford
economist, suggests using wrote a paper on this phenomenon;
Benford’s law to detect fraud. mathematicians later referred to it Funnily, of the 20 data sets
1995 American mathematician as Benford’s law. that Benford collected, six
Ted Hill proves that Benford’s of the sample sizes have
law can be applied to Recurring pattern leading digit 1. Notice
Benford’s law is evident in many
statistical distributions. anything strange about that?
situations, from the lengths of rivers
Rachel Fewster
2009 Statistical analysis to share prices and mortality rates. Statistical ecologist, New Zealand
of the Iranian presidential Some types of data fit the law better
election results shows that than others. Naturally occurring
they do not conform to data that extends over several orders
Benford’s law, suggesting of magnitude, from hundreds to
that the election may have millions, for example, fulfils the
been rigged.
See also: The Fibonacci sequence 106–11 ■ Logarithms 138–41
■ Probability 162–65 ■ Normal distribution 192–93
 MODERN MATHEMATICS 291

A BLUEPRINT
FOR THE
DIGITAL AGE
INFORMATION THEORY

I
n 1948, Claude Shannon, an
IN CONTEXT American mathematician and
electronics engineer, published
KEY FIGURE
a paper called A Mathematical
Claude Shannon (1916–2001)
Theory of Communication. This
FIELD launched the information age by
Computer science unlocking the mathematics of
information and showing how it
BEFORE could be transmitted digitally.
1679 Gottfried Leibniz At the time, messages could
develops the ancient idea only be transmitted using a
of binary numbering. continuous, analogue signal.
The main drawback to this was
1854 George Boole introduces Shannon demonstrates Theseus,
that waves become weaker the
the algebra that will form the his electromechanical “mouse”, which
further they travel, and increasing used a “brain” of telephone relays to
basis for computing. background interference creeps in. find its way around a maze.
1877 Austrian physicist Eventually, this “white noise”
Ludwig Boltzman develops overwhelms the original message.
the link between entropy Shannon’s solution was to divide Although Shannon was not the first
(measure of randomness) information into the smallest to send information digitally, he
and probability. possible chunks, or “bits” (binary fine-tuned the technique. For him,
digits). The message is converted it was not simply about solving the
1928 In the US, Ralph into a code made of 0s and 1s – technical problems of transmitting
Hartley, an electronics every 0 is a low voltage and every information efficiently. By showing
engineer, sees information 1 is a high voltage. In creating this that information could be expressed
as a measurable quantity. code, Shannon drew on binary as binary digits, he launched the
mathematics, the idea that figures theory of information – with
AFTER can be represented by just 0s and implications stretching into every
1961 German physicist Rolf 1s, which had been developed by field of science, and into every home
Landauer shows that the Gottfried Leibniz. or office with a computer. ■
manipulation of information
increases entropy. See also: Calculus 168–75 ■ Binary numbers 176–77 ■ Boolean algebra 242–47
292

WE ARE ALL JUST

SIX STEPS AWAY
FROM EACH OTHER
SIX DEGREES OF SEPARATION

IN CONTEXT Most individuals have a range of connections to people

KEY FIGURE from different parts of their lives.
Michael Gurevitch
(1930–2008)
FIELD
Number theory
These connections, in turn, have links to
BEFORE other groups and networks of people.
1929 Hungarian writer Frigyes
Karinthy coins the phrase “six
degrees of separation”.
AFTER
Further links to individuals who are three steps removed (for
1967 American sociologist example, a friend of a friend of a friend) reveal a wide range of
Stanley Milgram designs a people connected to each other.
“small world experiment” to
investigate people’s degrees of
separation and connectedness.
1979 Manfred Kochen of IBM Studies have suggested that, when connected
and Ithiel de Sola Pool at by our social networks, we are all, on average, six
MIT publish a mathematical steps away from each other.
analysis of social networks.
1998 In the US, sociologist

N
Duncan J. Watts and etworks are used to separation” social network diagram,
mathematician Steven model relationships which measures how connected
Strogatz produce the between objects or people people are to each other.
Watts–Strogatz random in many disciplines, including In 1961, Michael Gurevitch, an
graph model to measure computer science, particle physics, American postgraduate student,
connectedness. economics, cryptography, biology, published a landmark study of the
sociology, and climatology. One nature of social networks. In 1967,
type of network is a “six degrees of Stanley Milgram studied how many
 MODERN MATHEMATICS 293
See also: Logarithms 138–41 ■ Graph theory 194–95 ■ Topology 256–59 ■ The birth of modern statistics 268–71
■ The Turing machine 284–89 ■ Social mathematics 304 ■ Cryptography 314–17

degree of separation from other

published mathematicians. Erdős’s
coauthors had an Erdős number
of 1, anyone who had worked with
one of his coauthors had an Erdős
number of 2, and so on. This
concept captured the public’s
imagination following an interview
with American actor Kevin Bacon,
in which he said he had worked with
every actor in Hollywood or with
A 1 2 3 4 5 6 B
someone who had worked with
them. The term “Bacon number”
was coined to indicate an actor’s
degree of separation from Bacon.
In rock music, connections to
members of the heavy metal group
Black Sabbath are indicated by the
“Sabbath number”. To filter out
the truly well-connected, there
is the Erdős-Bacon-Sabbath
number (the sum of someone’s
The six degrees of separation theory shows how any two seemingly Erdős, Bacon, and Sabbath
unconnected people can be connected in no more than six steps by their friends
and acquaintances. This number may decrease with the growth of social media. numbers). Only a few individuals
have single-digit EBS numbers.
In 2008, Microsoft conducted
intermediate acquaintance links phrase “six degrees of separation”. research to show that everyone
were needed to connect strangers in Mathematicians have since tried on Earth is separated from every
the US. He had people in Nebraska to model the average degree of other person by only 6.6 people on
send a letter intended to eventually separation. Duncan Watts and average. As social media brings
reach a specific (random) person in Steven Strogatz showed that if you us ever closer, this number may
Massachusetts. Each recipient then have a random network with N reduce even further. ■
sent the letter on to a person they nodes, each of which has K links to
knew to get it closer to its target other nodes, then the average path
destination. Milgram studied how length between two nodes is ln N
many people each of the letters went divided by ln K (where ln means the
through to reach their targets. On natural logarithm). If there are 10
average, the letters that reached the nodes, each with four connections
target needed six intermediaries. to other nodes, then the average It is my hope that Six Degrees
This “small world theory” distance between two nodes chosen [a philanthropic project] will…
predated Milgram. In a 1929 short at random will be ln 10/ln 4 ≈ 1.66. [bring] a social conscience to
story Chains, Frigyes Karinthy social networking.
suggested that people’s average Other social networks Kevin Bacon
connection-number across the In the 1980s, friends of Hungarian
world might be six when the mathematician Paul Erdős, who
connecting factor is friendship. was well known for working
Karinthy, who was a writer, not collaboratively, coined the term
a mathematician, coined the “Erdős number” to indicate his
APOSITIVE
SMALLVIBRATION
CAN CHANGE
THE ENTIRE
THE BUTTERFLY EFFECT
COSMOS
296 THE BUTTERFLY EFFECT

IN CONTEXT
KEY FIGURE
Edward Lorenz (1917–2008)
FIELD
Probability
BEFORE
1814 Pierre-Simon Laplace
ponders the consequences of a
deterministic universe where
knowing all present conditions
can be used to predict the
future for all eternity.
1890 Henri Poincaré shows
there is no general solution to

I
the three-body problem, which n 1972, Edward Lorenz, an The idea that a butterfly flapping
predicts the motion of three American meteorologist and its wings in one part of the world
celestial bodies kept in contact mathematician, delivered could alter atmospheric conditions and
by gravity. Mostly, the bodies eventually produce a tornado elsewhere
a talk titled “Does the flap of a
has captured the popular imagination.
do not move in rhythmic, butterfly’s wings in Brazil set off
repeating patterns. a tornado in Texas?” This was the
origin of the term “butterfly effect”, then the tornado or other weather
AFTER which refers to the idea that a event would not have occurred
1975 Benoit Mandelbrot uses tiny change in atmospheric at all, or would have struck some
computer graphics to create conditions (which could be caused place other than Texas.
more complex fractals (shapes by anything, not just a butterfly) The title of the lecture was
that self-repeat). The Lorenz is enough to alter weather patterns not chosen by Lorenz himself,
attractor, which revealed the somewhere else in the future. If the but by physicist Philip Merilees,
butterfly effect, is a fractal. butterfly had not made its small the convener of the American
contribution to the initial conditions, Association for the Advancement

Edward Lorenz Born in 1917, in West Hartford, In developing a non-linear

Connecticut, Edward Lorenz model of the atmosphere, Lorenz
studied mathematics at Dartford stumbled across the area of
College and Harvard University, chaos theory that would later
gaining a masters degree at be dubbed the butterfly effect.
Harvard in 1940. After training He showed that even the most
as a meteorologist, he served powerful computers could not
with the US Army Air Corps in produce accurate long-term
World War II. After the war, weather forecasts. Lorenz
Lorenz studied meteorology at remained physically and
the Massachusetts Institute of mentally active until just
Technology and began to develop before his death in 2008.
ways to predict the behaviour of
the atmosphere. At that time, Key work
meteorologists used linear
statistical modelling to forecast 1963 Deterministic Nonperiodic
weather, and they often failed. Flow
 MODERN MATHEMATICS 297
See also: The problem of maxima 142–43 ■ Probability 162–65 ■ Calculus 168–75 ■ Newton’s laws of motion 182–83
■ Laplace’s demon 218–19 ■ Topology 256–59 ■ Fractals 306–11

Lorenz began investigating climate

modelling in the 1950s. By the early
1960s, he was attracting attention
for the unexpected results of a toy
climate model (“toy” meaning that
A butterfly flaps its wings in it was a simplistic model made to
the Amazonian jungle, and demonstrate processes concisely). The amazing thing is
subsequently a storm The model predicted the way the that chaotic systems don’t
ravages half of Europe. atmosphere would evolve in terms always stay chaotic.
Terry Pratchett and of three data points, such as air Connie Willis
Neil Gaiman pressure, temperature, and wind American writer
British authors speed. Lorenz found that the results
were chaotic. He compared two
sets of results, each starting with
near-identical sets of data, noting
that the atmospheric conditions
developed along near-identical
lines at first, but then changed in
of Science’s annual meeting in completely different ways. He also values on the x, y, and z axes
Boston. Lorenz had been late to found that while every starting represented, for example, air
provide information about his point in his model rendered unique temperature, pressure, and
proposed talk, so Merilees had results, they were all confined humidity (or triplets of other
improvised, basing his choice of within certain limits. weather data). In 1963, when it
words on what he knew of Lorenz’s became possible to plot this data,
work and an earlier comment that Strange attractor the shape created became known
“one flap of a seagull’s wings” The computing power available as the Lorenz attractor. Each
could be enough to change the to Lorenz in the early 1960s was starting point evolves into a looping
weather forecast. unable to plot the modelled line that swings from one quadrant
atmospheric variables in a three- of the space to another – indicating,
Chaos theory dimensional space, where the for example, a change from wet ❯❯
The butterfly effect is a popular
introduction to chaos theory, which
looks at the way complex systems
are highly sensitive to initial
conditions and are thus extremely
unpredictable. Chaos theory has
practical relevance to areas such
as population dynamics, chemical
engineering, and financial markets,
and helps in the development of
artificial intelligence.

In a Lorenz attractor, small changes

in starting conditions result in huge
changes to the paths each line takes,
yet the lines still fall within the confines
of the same shape, providing order
within the chaos.
298 THE BUTTERFLY EFFECT

Many dynamic systems of nature appear deterministic,

following laws that always apply.

Chaos: when the present

determines the future, but
the approximate present
does not approximately If we know the initial conditions very precisely, we can
determine the future. determine the future conditions exactly.
Edward Lorenz

However, the system is highly sensitive. A small change

in the initial conditions will lead to a big difference
in the future conditions.
and windy weather to hot, dry
conditions, and all states in
between. Each starting point
leads to a unique evolution, but
all the lines, whatever the start If we only know an approximation of initial conditions, our
point, fall into the same region predictions will be imprecise.
of the space. After many iterations,
run for long periods, that region
becomes a beautiful looping
surface. The individual lines
within the attractor are highly
unstable in their trajectories; The system is chaotic.
those that start in the same area
often move far apart at a later
point, and lines with very different
starting points may end up tracking planets sweep through space as acting upon it are known, it is
each other closely for long periods. they orbit the Sun; and Isaac possible to determine its future
However, the attractor shows that Newton combined this knowledge location and condition with
as a whole, the system is stable. with physical laws covering gravity perfect accuracy.
There is no possible starting point and motion. Along with Gottfried
within the attractor that can lead Leibniz, Newton is credited with The three-body problem
to a trajectory that escapes from developing calculus, a system of Nevertheless, Newton found a flaw
it. This apparent contradiction is mathematics designed to analyse with this deterministic view of the
at the heart of chaos theory. and predict the behaviours of more Universe. He reported difficulties
complex systems. Using calculus, in analysing the movements of
Finding the right path the relationships between any three bodies bound together by
The roots of chaos theory lie in complex variables can – in theory – gravity – even when those bodies
early attempts to understand be predicted by solving a particular were as seemingly stable as the
and predict motion, especially of differential equation. Earth, Moon, and Sun. Later
heavenly bodies. For example, in These physical laws and attempts to analyse the movement
the 17th century, Galileo formulated analytical tools can demonstrate of the Moon to improve navigation
laws about the way pendulums that the Universe is deterministic – were plagued by inaccuracies.
swing and how objects fall; if the exact location and condition In 1890, French mathematician
Johannes Kepler showed how of an object and all the forces Henri Poincaré showed that there
 MODERN MATHEMATICS 299
was no generalized, predictable pendulum (one that is losing energy
way in which three bodies move because of friction), the oscillatory
around each other. In a few cases, motion will diminish until the
where the bodies start in very imaginary body reaches the fixed
specific places, the motion is point – when it stops moving.
periodic – it repeats the same When considering the motion Determinism was equated
paths over and over again. Mostly, of an imaginary body with respect with predictability before
Poincaré argued, the three bodies to several others, the geodesic path Lorenz. After Lorenz, we came
do not retrace their paths, and their becomes very complicated. If it to see that… in the long run,
movement is called aperiodic. were possible to set the start things could be unpredictable.
Mathematicians hoping to solve conditions precisely, it would be Stephen Strogatz
this “three-body problem” have possible to create every conceivable American mathematician
abstracted it to consider imaginary path. Some would be periodic,
bodies moving around surfaces and repeating a path of whatever
spaces with specific curvature. The complexity over and over again.
curvature of an imaginary body can Others would be unstable initially
be a mathematical representation of but would settle into a limit cycle
the forces (such as gravity) acting eventually. A third kind would fly
on it. The path the imaginary body off to infinity – perhaps straight about the starting conditions.
takes in each case is called the away, or perhaps after a period of This is the essence of chaos
geodesic path (see below). In a apparent stability. theory. Even though the system is
simple case, such as the movement deterministic, every measurement
of a pendulum or the orbit of a Approximations of that system is an approximation.
planet around a star, this imaginary Although it has been studied by Therefore, any mathematical
body oscillates (moves back and physicists and mathematicians model based on those uncertain
forth) around a fixed point on the alike, the three-body problem is measurements will very possibly
surface, following a repeating path largely theoretical. When it comes develop in a different way from the
and creating what is called a limit to a real physical system, there is real thing. Even a small uncertainty
cycle. In the case of a damped no way to be absolutely precise is enough to create chaos. ■

The geodesic path of a planet

The gravity
The geodesic path of well of a star
a planet orbiting a star
in a predictable way is
shown in the left-hand
image. The image on
the right shows how the
presence of three other
celestial bodies – perhaps
nearby planets or other
stars – complicates the
planet’s path, making it
unpredictable, or chaotic.

Three bodies
A planet with exert gravitational
no neighbouring effects on the planet.
bodies The geodesic path
The geodesic path of the planet is scrambled
of the planet forms by the proximity of the
a predictable shape. three bodies.
300

LOGICALLY THINGS
CAN ONLY PARTLY
BE TRUE
FUZZY LOGIC

T
he binary logic of any developed in 1965 by Lotfi Zadeh,
IN CONTEXT computer is clear: given an Iranian–American computer
valid inputs, it will provide scientist. Zadeh claimed that as a
KEY FIGURE
appropriate outputs. However, system becomes more complex,
Lotfi Zadeh (1921–2017)
binary computer systems are precise statements about it become
FIELD not always well suited for dealing meaningless; the only meaningful
Logic with real-world inputs that are statements about it are imprecise.
ambiguous or unclear. In the case Such situations demand a many-
BEFORE of handwriting recognition, for valued (fuzzy) reasoning system.
350 bce Aristotle develops a example, a binary system would Standard set theory allows an
system of logic that dominates not be sufficiently subtle. A system element to either belong or not
Western scientific reasoning controlled by fuzzy logic, however, belong to a set, but fuzzy set theory
until the 19th century. allows for degrees of truth that can allows degrees of membership or
better analyse complex phenomena, a continuum. Similarly, fuzzy logic
1847 George Boole invents
including human actions and allows a range of truth values for a
a form of algebra in which thought processes. Fuzzy logic is proposition – not just completely
variables can have one of an offshoot of the fuzzy set theory true or completely false, the two
only two values (true or false), values of Boolean logic. Fuzzy
paving the way for symbolic, truth values also require fuzzy
mathematical logic. logical operators – for example, the
1930 Polish logicians Jan fuzzy version of the AND operator
Łukasiewiecz and Alfred of Boolean algebra is the MIN
operator, which outputs the
Tarski define a logic with The classes of objects minimum of the two inputs.
infinitely many truth values. encountered in the real
AFTER physical world do not have Creating fuzzy sets
1980s Japanese electronics precisely defined A basic computer program that
companies use fuzzy logic criteria of membership. mimics the simple human task of
control systems in industrial Lotfi Zadeh soft-boiling an egg might apply a
and domestic appliances. single rule: boil the egg for five
minutes. A more sophisticated
program would, like a human,
take the weight of the egg into
account. It might divide eggs
 MODERN MATHEMATICS 301
See also: Syllogistic logic 50–51 ■ Binary numbers 176–77 ■ Boolean algebra 242–47 ■ Venn diagrams 254 ■ The logic
of mathematics 272–73 ■ The Turing machine 284–89

BOOLEAN LOGIC FUZZY LOGIC

Is it hot outside? Is it hot outside?

Yes: 1 No: 0 Very hot: 1 Quite hot: 0.75 Average: 0.5 A bit cold: 0.25 Freezing: 0

Fuzzy logic recognizes a continuum of truth values instead of the Boolean binary values
of “yes” (1) or “no” (0). These fuzzy values resemble probabilities, but are fundamentally
quite distinct – they indicate the degree to which a proposition is true, not how likely it is.

into two sets – small eggs of 50 g general approach. The first step is deduce that an 80 g egg should
(1.76 oz) or less, and large ones to make the data fuzzy – every egg be boiled for both four and six
over 50 g – and boil the former for is regarded as both large and small, minutes (with degrees of almost 0
four minutes, and the latter for six. belonging to both sets to different and almost 1 respectively). This
Fuzzy logicians call these crisp degrees. For example, a 50 g egg output is then defuzzified to give a
sets: each egg either does or does would have a membership degree crisp logical output that can be used
not belong. of 0.5 for both sets, while an 80 g by the control system. As a result,
To achieve a perfectly cooked (2.82 oz) egg would be “large” with the 80g egg would be assigned a
egg, however, the boiling time degree nearly 1, and also “small” boiling time of nearly 6 minutes.
must be adjusted to match the with degree nearly 0. A fuzzy rule is Fuzzy logic is now a ubiquitous
weight of the egg. While an then applied, with large eggs boiled part of computer-controlled
algorithm could use traditional for six minutes and small eggs for systems. It has many applications,
logic to divide a set of eggs into four. Through a process called fuzzy from forecasting weather to
precise weight ranges and assign inference, the algorithm applies the trading stocks, and plays a vital
exact cooking times, fuzzy logic rule to each egg based on its fuzzy role in programming artificial
achieves this result with a more set membership. The system will intelligence systems. ■

Artificial intelligence and error, and expert systems,

in which the AI draws upon a
Fuzzy control systems can work database of knowledge provided
effectively with uncertainties by human programmers, have
in the everyday world, and greatly extended the abilities
are therefore used in artificial of AI. Nevertheless most AI is
intelligence (AI) systems. The “narrow”, in that it is tasked
fuzziness of AI helps to give with doing one job very well,
the illusion of a self-directing generally better than a human
intelligence, but in reality fuzzy can, but it cannot learn to do
logic processes data to smooth anything else and is unaware
out uncertainty. AI is therefore of what it does not know. A
entirely the product of a pre- general AI that can direct its
A humanoid robot using AI works programmed set of rules. own learning in the same way
at the front desk of a Henn-na hotel Techniques such as machine as evolved intelligence (such as
in Tokyo, which claims to be the learning, in which AIs program human intelligence) is the next
world’s first hotel with robotic staff. themselves by a process of trial goal of computer science.
302

A GRAND UNIFYING
THEORY OF
MATHEMATICS
THE LANGLANDS PROGRAM

I
n 1967, the young Canadian– seem fundamentally different:
IN CONTEXT American mathematician while sine waves are continuous,
Robert Langlands suggested a integers are discrete.
KEY FIGURE
set of profound links between two
Robert Langlands (1936–)
major and seemingly unconnected Langlands’ letter
FIELD areas of mathematics – number In a 17-page handwritten letter
Number theory theory and harmonic analysis. to number theorist André Weil in
Number theory is the mathematics 1967, Langlands offered several
BEFORE of integers, in particular prime conjectures linking number theory
1796 Carl Gauss proves the numbers. Harmonic analysis (in and harmonic analysis. Realizing
quadratic reciprocity theorem, which Langlands specialized) is the its significance, Weil had the letter
relating the solvability of mathematical study of waveforms, typed up and circulated among
quadratic equations to exploring how they can be broken number theorists through the late
prime numbers. down to sine waves. These fields 1960s and ’70s. Once they had
1880–84 Henri Poincaré
develops the concept of
automorphic forms – tools Number theory deals Harmonic analysis
that allow us to keep track with the properties analyses complicated
of complicated groups. of and relationship functions, breaking them
between integers. into groups of sine waves.
1927 Austrian mathematician
Emil Artin extends the
reciprocity theorem to groups.
AFTER
The Langlands Program joins together these
1994 Andrew Wiles uses a
seemingly disparate branches of mathematics.
special case of Langlands’
conjectures to translate
Fermat’s last theorem from
a problem in number theory
to one in geometry, enabling The Program can be described as “a grand
him to solve it. unifying theory of mathematics”.
 MODERN MATHEMATICS 303
See also: Fourier analysis 216–17 ■ Elliptic functions 226–27 ■ Group theory 230–33 ■ The prime number theorem 260–61
■Emmy Noether and abstract algebra 280–81 ■ Proving Fermat’s last theorem 320–23

been made public, Langlands’

conjectures became influential
across mathematics, and continue
to shape research 50 years later.

Uncovering links +4
Langlands’ ideas involve highly
technical mathematics. In basic
terms, his areas of interest are
Galois groups and functions called
automorphic forms. Galois groups
turn up in number theory and are
a generalization of the groups that
Évariste Galois used in order to Modular (“clock”) arithmetic involves number systems with finite
study roots of polynomials. sets of numbers. On a 12-hour clock, for example, if you count on four
hours from 10 o’clock, you get 2 o’clock; 10 + 4 = 2, because the
Langlands’ conjectures are
remainder of 14  12 is 2. In the Langlands program, numbers are
significant in that they allowed usually manipulated by modular arithmetic.
problems from number theory to
be reframed in the language of
harmonic analysis. The Langlands Langlands’ marriage of harmonic at the heart of mathematics. In the
Program has been described as analysis and number theory could 1980s, Ukrainian mathematician
a mathematical Rosetta Stone, lead to a wealth of new tools, just Vladimir Drinfel’d expanded the
helping to translate ideas from as the 19th-century unification of Program’s scope to show that
one area of mathematics into electricity and magnetism into there might be a Langlands-type
another. Langlands himself has electromagnetism provided a new connection between specific topics
helped to develop the means for understanding of the physical world. within harmonic analysis and
working on the Program, including By finding new links between others within geometry. In 1994,
generalizing functoriality – a way mathematical fields that seem Andrew Wiles used one of
of comparing the structures of profoundly different, the Program Langlands’ conjectures to help
different groups. has revealed some of the structures solve Fermat’s last theorem. ■

Robert Langlands Born near Vancouver, Canada, in Langlands began studying the
1936, Robert Langlands did not relationship between integers
plan to go to university until a and periodic functions as part of
teacher “took up an hour of class research into patterns in prime
time” to publicly implore him to numbers. He was awarded the
make use of his talents. He was Abel Prize in 2018 for his
also a gifted linguist, but at 16, “visionary” Program.
he enrolled at the University of
British Columbia, Canada, to study
mathematics. He later moved to Key works
the US, where he was awarded a
doctorate from Yale University in 1967 Euler Products
1960. Langlands taught at 1967 Letter to André Weil
Princeton, Berkeley, and Yale 1976 On the Functional
before moving to the Institute for Equations Satisfied by
Advanced Study (IAS), where he Eisenstein Series
still occupies Einstein’s old office. 2004 Beyond Endoscopy
304

ANOTHER ROOF,
ANOTHER PROOF
SOCIAL MATHEMATICS

H
ungarian mathematician number of 2, and so on. Albert
IN CONTEXT Paul Erdős wrote and Einstein has an Erdős number
cowrote around 1,500 of 2; Paul Erdős’s number is 0.
KEY FIGURE
academic papers in his lifetime. Oakland University runs the
Paul Erdős (1913–96)
He worked with more than 500 Erdős Number Project, which
FIELD others in the global mathematical analyses collaboration among
Number theory community across different research mathematicians. The
branches of mathematics, including average Erdős number is around
BEFORE number theory (the study of 5. The rarity of an Erdős number
1929 Hungarian author integers) and combinatorics – a higher than 10 indicates the
Frigyes Karinthy postulates field of mathematics concerned degree of collaboration within
the concept of six degrees with the number of permutations the mathematical community. ■
of separation in his short that are possible in a collection
story, Láncszemek (Chains). of objects. His motto, “Another
roof, another proof”, referred to
1967 American social his habit of staying at the homes
psychologist Stanley Milgram of fellow mathematicians in order
conducts experiments on to “collaborate” for a while.
the interconnectedness The Erdős number, first used Erdős has an amazing ability
of social networks. in 1971, indicates how far a to match problems with
AFTER mathematician is removed from people. Which is why
1996 The Bacon number is Erdős in their published work. so many mathematicians
introduced on an American TV
To qualify for an Erdős number, benefit from his presence.
show. It indicates the number
a person has to have written a Béla Bollobás
mathematical paper – someone Hungarian–British mathematician
of degrees of separation an who coauthored a paper with Erdős
actor has from American actor would have an Erdős number of 1.
Kevin Bacon. Someone who worked with a
2008 Microsoft conducts coauthor (but not with Erdős
the first experimental study directly) would have an Erdős
into the effects of social
media on connectedness. See also: Diophantine equations 80–81 ■ Euler’s number 186–91
■ Six degrees of separation 292–93 ■ Proving Fermat’s last theorem 320–23
 MODERN MATHEMATICS 305

PENTAGONS ARE
JUST NICE TO
LOOK AT
THE PENROSE TILE

T
ile patterns have been Pentagon
IN CONTEXT a feature of art and
Star
construction for millennia,
KEY FIGURE
especially in the Islamic world. The
Roger Penrose (1931–)
need to fill two-dimensional space
FIELD as efficiently as possible led to the
Applied geometry study of tessellations – the fitting
together of polygons with no gaps
BEFORE or overlap. Some natural structures,
4000 bce Sumerian buildings such as a honeycomb, tessellate.
incorporate tessellations into There are three regular shapes
wall decorations. that tessellate on their own,
without the need for another shape: Kite Dart
1619 Johannes Kepler
the square, equilateral triangle, and
conducts the first documented regular hexagon. However, many Penrose tiling consists of kites
study of tessellations. irregular shapes also tessellate, and and darts, producing a nonperiodic
1891 Russian crystallographer semiregular tessellations involve tessellation. Shapes with five-fold
more than one regular shape. The symmetry, such as pentagons and
Evgraf Fyodorov proves there stars, can also be identified.
are only 17 possible groups pattern of such tessellations usually
that form periodic tilings repeats. This is known as a
of the plane. “periodic tessellation”. he created tiles using kite and dart
Nonperiodic tessellations, in shapes. The kite and dart must be
AFTER which the pattern does not repeat, exactly the same shape as the ones
1981 Dutch mathematician are harder to find, although some shown (above); the area of the kite
Nicolaas Govert de Bruijn regular shapes can be combined to that of the dart is expressed by
explains how to construct to create nonperiodic tessellations. the golden ratio. Although no part
Penrose tilings from five British mathematician Roger of the tiling matches another part
families of parallel lines. Penrose investigated whether exactly, the pattern does repeat on
any polygons could only lead to a larger scale in a similar way
1982 Israeli engineer Dan nonperiodic tessellations. In 1974, to a fractal. ■
Shechtman discovers quasi-
crystals whose structure is See also: The golden ratio 118–23 ■ The problem of maxima 142–43
similar to Penrose tilings. ■ Fractals 306–11
ENDLESS
VARIETY AND
UNLIMITED
COMPLICATION
FRACTALS
308 FRACTALS
IN CONTEXT
KEY FIGURE
Benoit Mandelbrot
(1924–2010)
FIELDS
Geometry, topology
BEFORE
c. 4th century bce Euclid
sets out the foundations of
geometry in Elements.
AFTER
1999 The study of “allometric
scaling” applies fractal growth
to metabolic processes within
biological systems, leading to
valuable medical applications.
2012 In Australia, the largest
3-D map of the sky suggests

A
that the Universe is fractal up
fter Euclid, scholars and A computer graphic shows a
to a point, with clusters of
mathematicians modelled fractal pattern derived from the
matter within larger clusters, Mandelbrot set. Mesmerizingly
the world in terms of
but ultimately matter is beautiful, such images produced
simple geometry: curves and
distributed evenly. with fractal-generating software
straight lines; the circle, ellipse, make popular screen savers.
2015 Fractal analysis is and polygons; and the five Platonic
applied to electrical power solids – the cube, the tetrahedron,
networks, leading to the the octahedron, the dodecahedron, findings of earlier mathematicians.
modelling of the frequency and the icosahedron. For much of In 1872, German mathematician
of power failure. the past 2,000 years, the prevailing Karl Weierstrass had formalized
assumption has been that most the mathematical concept of
natural objects – mountains, trees, “continuous function”, meaning
and so on – can be deconstructed that changes in the input result
into combinations of these shapes in roughly equal changes in the
to ascertain their size. However, in output. Composed entirely of
1975, Polish-born mathematician corners, the Weierstrass function
Benoit Mandelbrot drew attention has no smoothness anywhere,
A geometry able to include to fractals – non-uniform shapes however much it is magnified.
mountains and clouds now that echo larger and smaller shapes This was seen at the time as a
exists… Like everything in a structure such as a jagged mathematical abnormality that,
in science this new mountaintop. Fractals, a word unlike the sensible Euclidean
geometry has very, very derived from the Latin fractus, shapes, had no real-world relevance.
deep and long roots. meaning “broken”, would eventually In 1883, another German
Benoit Mandelbrot lead to the topic of fractal geometry. mathematician, Georg Cantor, built
on work by British mathematician
A new geometry Henry Smith to demonstrate how
Although it was Mandelbrot who to create a line that is nowhere
brought fractals to the attention of continuous and has zero length. He
the world, he was building on the did so by drawing a straight line,
 MODERN MATHEMATICS 309
See also: The Platonic solids 48–49 ■ Euclid’s Elements 52–57 ■ The complex
plane 214–15 ■ Non-Euclidean geometries 228–29 ■ Topology 256–59

removing the middle third (leaving In 1918, German mathematician

two lines and a gap), and then Felix Hausdorff proposed the
repeating the process ad infinitum. existence of fractional dimensions.
The result is a line composed Whereas the simple line, plane, and
entirely of disconnected points. solid occupy one, two, and three
Like the Weierstrass function, this dimensions respectively, these
“Cantor set” was considered new shapes could be given non-
unsettling by the mathematical whole-number dimensions. For
establishment, who branded example, the British coastline could,
these new shapes “pathological” – in theory, be measured with a
meaning “lacking usual properties”. one-dimensional rope, but inlets Benoit Mandelbrot
In 1904, Swedish mathematician would require string, and crevices
Helge von Koch constructed a require thread. This implies that the Born into a Jewish family
shape known as the Koch curve or coastline cannot be measured in in Warsaw in 1924, Benoit
“Koch snowflake”, which repeated one dimension. The British coastline Mandelbrot left Poland in 1936
a triangular motif at an ever smaller has a Hausdorff dimension of 1.26, to escape the Nazis. His family
size. This was followed in 1916 like the Koch curve. went first to Paris and then
by the Sierpinski triangle, or to the south of France. After
Sierpinski gasket, composed Dynamic self-similarities World War II, Mandelbrot
entirely of triangular holes. French mathematician Henri gained scholarships to study in
All these shapes possess self- Poincaré found that dynamical France and then the US, before
similarity, which is a key property systems (systems that change over returning to Paris, where he
was awarded a doctorate in
of fractal geometry. This means time) also had fractal properties of
mathematical sciences from
that enlargement of a portion of self-similarity. By their nature,
the city’s university in 1952.
the shape reveals smaller replicas dynamical states are “non- In 1958, Mandelbrot joined
with equal detail. Mathematicians deterministic”: two systems that IBM in New York, where his
realized that this was a fundamental are nearly identical can lead to very role as a researcher gave him
property of natural growth – a different behaviours even when the the space and facilities to
repetition of a pattern on many initial conditions are also almost develop new ideas. In 1975,
scales, from the macro to the micro. identical. This phenomenon is ❯❯ he coined the term “fractal”,
and in 1980 he unveiled the
Mandelbrot set, a structure
that became synonymous with
the new science of fractal
The Mandelbrot set Its boundary is highly
geometry. The topic gained
has an extraordinarily complex and infinitely
popular appeal in 1982 with
elaborate structure. squiggly.
the publication of his book
The Fractal Geometry of
Nature. Mandelbrot received
many honours and prizes for
Magnifying any part of it, however small, reveals a his work, including France’s
replica of the set itself. Légion d’honneur in 1989. He
died in 2010.

Key works
No one can fully comprehend the set’s
1982 The Fractal Geometry
endless variety and unlimited complication. of Nature
310 FRACTALS
Timeline of fractals
1883
Cantor set
Constructed by repeatedly removing
the middle third from a succession
of lines, the Cantor set creates
a series of intervals.

1872 1904
Weierstrass function Koch snowflake
Composed of corners, the Weierstrass The shape becomes
function will never appear smooth infinitely more intricate as
however much it is magnified. more triangles are added.

popularly known as the “butterfly letter standing for a fixed number) available at the time, Julia and
effect”, after the frequently cited to it, and then repeating the Fatou were unable to see the true
example of the massive effect a process, some initial values would significance of their discovery, but
single butterfly can theoretically diverge to infinity while others they had found what would become
have on a weather system when would converge to a finite value. known as the Julia set.
it causes a small disturbance by Julia and Fatou mapped these
flapping its wings. The differential different values on a complex The Mandelbrot set
equations devised by Poincaré plane, noting which ones converged In the late 1970s, Benoit Mandelbrot
to prove his theory implied the and which ones diverged. The used the term “fractal” for the first
existence of dynamical states that boundaries between these regions time. Mandelbrot had become
possess self-similarity much like were self-replicating, or fractal. With interested in the work of Julia and
fractal structures. Large-scale the limited computational power Fatou while working at the IT
weather systems, such as major company IBM. With the computer
cyclonic flows, for instance, repeat facilities available at IBM, he was
themselves on much smaller scales, able to analyse the Julia set in great
right down to gusts of wind. detail, noting that some values of
In 1918, French mathematician the constant (c) gave “connected”
Gaston Julia, a former student of sets, in which each of the points is
Poincaré, explored the concept of joined to another, and others were
self-similarity when he began to disconnected. Mandelbrot mapped
map the complex plane (the each value of c on the complex
coordinate system based on plane, colouring the connected
complex numbers ) under a process sets and the disconnected sets
called iteration – entering a value in different colours. This led, in
into a function, obtaining an output, 1980, to the creation of the
and then plugging that back into Mandelbrot set.
the function. Along with George
Fatou, who undertook similar
Infinite complexity is suggested
research independently, Julia by the self-similarities of a Romanesco
found that by taking a complex cauliflower. The natural world is full
number, squaring it, adding a of fractals, from ferns and sunflowers
constant (a fixed number or a to ammonites and seashells.
 MODERN MATHEMATICS 311

1916 1980
Sierpinski triangle Mandelbrot set
Repeatedly adding triangles Infinitely squiggly, the Mandelbrot
within triangles creates an set looks more elaborate the more it
infinitely lacy pattern. is magnified.

1918
Julia set
The Julia set, which examined
dynamical systems, displays regular
and chaotic iterations.

Beautifully complex, the Mandelbrot Many natural objects exhibit self- viruses and the development of
set displays self-similarity at all similarity, including mountains, tumours. They are also used in
scales: magnification reveals rivers, coastlines, clouds, weather engineering, particularly in the
smaller replicas of the Mandelbrot systems, blood circulatory development of polymer and
set itself. In 1991, Japanese systems, and even cauliflowers. ceramic materials. The structure
mathematician Mitsuhiro Shishikura Being able to model these diverse and evolution of the Universe can
proved that the boundary of the phenomena using fractal also be modelled on fractals, as
Mandelbrot set has a Hausdorff geometry enables us to better can the fluctuations of economic
dimension of 2. understand their behaviour and markets. As the range of
evolution, even if that behaviour applications grows, along with ever-
Application of fractals is not entirely deterministic. increasing computational capacity,
Fractal geometry has allowed Fractals have applications fractals are becoming integral to
mathematicians to describe the in medical research, such as our understanding of the seemingly
irregularity of the real world. understanding the behaviour of chaotic world in which we live. ■

Fractals and the arts

Self-similarity on infinite scales The work of the Japanese artist
is explored in philosophy and the Katsushika Hokusai, with its
arts, often to produce a meditative swirling repeated motifs, is often
effect. It is a key tenet of Buddhist cited as an example of fractal
meditation and mandalas (symbols use in art, as is the architecture
used in rituals to represent the of Catalan artist Antoni Gaudí.
Universe), and is also used to The musical “rave” scene
suggest the infinite nature of God in the US and UK in the late
in Islamic decoration, such as 1980s and early ’90s was
tilework. Self-similarity is even linked to a surge of interest
suggested in the poem “Auguries in fractal art. Nowadays there
Under the Wave off Kanagawa by of Innocence” by the 19th-century are many fractal-generating
Japanese artist Katsushika Hokusai British poet William Blake, which computer programs, making it
(1760–1849) employs the concept of begins with the line “To see a possible for the general public
self-similarity to dramatic effect. world in a grain of sand”. to create fractals.
312

FOUR COLOURS
BUT NO MORE
THE FOUR-COLOUR THEOREM

IN CONTEXT How many colours do you need to colour a map so that no two
KEY FIGURES countries with a common border have the same colour?
Kenneth Appel (1932–2013),
Wolfgang Haken (1928–)
FIELD
Topology It can’t be done with just two or three colours.
BEFORE
1852 South African law
student Francis Guthrie asserts
that four colours are needed to
colour a map so that adjacent In 1890, it is proved that In 1976, a computer is used
areas are not the same colour. any map can be coloured to prove that no more than
with five colours. four colours are necessary.
1890 British mathematician
Percy Heawood proves that
five colours are sufficient to
colour any map.
AFTER Four colours are enough
1997 In the US, Neil Robertson, to colour a map.
Daniel P. Sanders, Robin
Thomas, and Paul Seymour

C
provide a simpler proof
artographers have long map using only four colours.
of the four-colour theorem. known that any map, Mathematicians searched for a
2005 Microsoft researcher however complicated, proof for this deceptively simple
Georges Gonthier proves the can be coloured in with just four theorem for more than 120 years,
four-colour theorem with colours, so that no two nations or making it one of the most enduring
general purpose theorem- regions sharing a border are the unsolved theorems in mathematics.
proving software. same colour. Although five colours The first person to formulate the
can seem to be necessary, there is four-colour theorem is thought to
always a way of re-colouring the have been Francis Guthrie, a South
 MODERN MATHEMATICS 313
See also: Euler’s number 186–91 ■ Graph theory 194–95 ■ The complex plane 214–15 ■ Proving Fermat’s last
theorem 320–23

Any combination of shapes in a

plane, however complex the pattern, Computer proofs
can be coloured in using just four
colours so that no two adjacent When Appel and Haken
shapes have the same colour. proved the four-colour
theorem in 1977, it was the
first time that a computer
he had made a mistake that he had been used to prove a
could not rectify. Heawood did mathematical theorem. This
prove correctly that no more than was controversial among
five colours were needed to colour mathematicians, who were
any map. used to solving problems
Mathematicians continued to through logic that could
work on the problem, and gradual be checked by their peers.
progress was made. In 1922, Philip Appel and Haken had used
Franklin proved that any map with the computer to carry out a
25 regions or fewer was four- proof by exhaustion – all
colourable. The figure of 25 was possibilities were meticulously
slowly increased; Norwegian checked one by one, a feat
that would have been
African law student. He had mathematician Øystein Ore and
impossible to do manually.
coloured a map of the English American mathematician Joel
The question was whether
counties using just four colours Stemple together achieved 39 in a long calculation that could
and believed that the same could 1970, and Frenchman Jean Mayer not be checked by humans,
be done with any map, however lifted the figure to 95 in 1976. followed by a simple verdict
complex. In 1852, he asked his of “yes, the theorem has been
brother Frederick, who was New hope proved”, could be accepted.
studying under mathematician The introduction of supercomputers, Many mathematicians
Augustus De Morgan in London, computers capable of handling argued that it could not.
if his theory could be proved. huge amounts of data, in the 1970s Proof by computers remains
Admitting that he could not prove revived interest in solving the four- controversial, but advances
the theorem, De Morgan shared it colour theorem. Although German in technology have increased
with Irish mathematician William mathematician Heinrich Heesch confidence in their reliability.
Hamilton. Hamilton went on to suggested a method for doing this,
attempt to prove the theorem he did not have sufficient access
himself, but did not succeed. to a supercomputer to test it.
Wolfgang Haken, a former student
False start of Heesch’s, became interested in
In 1879, British mathematician the problem, and began to make
Alfred Kempe claimed a proof progress after meeting computer
for the four-colour theorem in programmer Kenneth Appel at the
the scientific journal Nature. University of Illinois in the USA.
Kempe received plaudits for this The pair finally cracked the
work, and two years later became a problem in 1977. Relying completely
Fellow of the Royal Society partly on computing power – the first
on the strength of his proof. proof in the history of mathematics The IBM System/370 computer
However, in 1890, fellow British to do so – they examined around c. 1970 was one of the first computers
to use virtual memory, a working
mathematician Percy Heawood 2,000 cases, involving billions of memory system that allowed it to
found a hole in Kempe’s proof, and calculations and using 1,200 hours process large amounts of data.
Kempe himself acknowledged that of computer time. ■
314
IN CONTEXT

SECURING DATA
KEY FIGURES
Ron Rivest (1947–),
Adi Shamir (1952–),

WITH A ONE-WAY
Leonard Adleman (1945–)
FIELD
Computer science

CALCULATION
BEFORE
9th century ce Al-Kindi
develops frequency analysis.
1640 Pierre de Fermat
CRYPTOGRAPHY states his “little theorem”
(on primality), which is still
used as a test when searching
for primes to use in public
key encryption.
AFTER
2004 Elliptic curves are first
used in cryptography; they use
smaller keys but offer the same
security as the RSA algorithm.
2009 An anonymous computer
scientist mines the first Bitcoin,
a cryptocurrency without a
central bank. All transactions
are encrypted but public.

C
ryptography is the
development of means
of secret communication.
It has become a ubiquitous feature
of modern life, with almost every
connection between one digital
device and another starting with a
“handshake”, in which the devices
agree on a way of securing their
connection. That handshake is
often the result of the work of three
mathematicians: Ron Rivest, Adi
Shamir, and Leonard Adleman.
In 1977, they developed the RSA
algorithm (named for their initials),
an encryption procedure that won
them the Turing Award in 2002.
The RSA algorithm is special
 MODERN MATHEMATICS 315
See also: Group theory 230–33 ■ The Riemann hypothesis 250–51 ■ The Turing machine 284–89 ■ Information
theory 291 ■ Proving Fermat’s last theorem 320–23

Data can contain sensitive

information which needs to be kept safe and secure
when it is sent.
The work did not really
need mathematics, but
mathematicians tended
to be good at it. Computer
calculations led to the Codes have been used
Joan Clarke for centuries, but some
British cryptanalyst creation of more
advanced codes. were easy to break.

These codes are Cryptography

because it ensures that any third almost irreversible enables data to be
party monitoring the connection without the correct “key” transmitted securely.
will be completely unable to figure to break them.
out any private details.
One main reason people have
needed to encrypt communications
is to ensure financial transactions cipher is an algorithm (a systematic, is known as the Caesar cipher
can happen without banking repeatable method) – in this case, (or Caesar shift) after the Roman
information falling into the wrong to substitute each letter with one dictator Julius Caesar, who used it
hands. However, encryption is used in another position in the alphabet. in the 1st century bce. The Caesar
against all kinds of third-party The key is +1, because each of the cipher is an example of symmetric
“adversary” – a rival company, an letters in plaintext is substituted encryption, as the same cipher
enemy power, or a security service. with the letter +1 along in the and key are used (in reverse) to
Cryptography is an ancient practice. alphabet. If the key were 6, then decipher the message.
Mesopotamian clay tablets from the cipher would turn the same
c. 1500 bce were often encrypted plaintext “HELLO” into “BZFFI”. Deciphering processes
to protect recipes for pottery glazes This simple substitution system Given enough paper and time, it
and other such commercially is relatively easy to figure out a
valuable information. Caesar cipher by trying out every
possible substitution. In modern
Cipher and key terms this is known as a “brute
The term “cryptography” comes force” technique. More complex
from the Greek for “hidden writing ciphers and keys make brute force
study”. For much of history it was more time-consuming – and, before
used to secure written messages. computers, effectively unworkable
The unencrypted message is for messages long enough to hold ❯❯
known as the plaintext, while the
encrypted version is called the
Cipher wheels, such as this British
ciphertext. For example, “HELLO” example from 1802, sped up the
might become “IFMMP”. Going decryption of Caesar ciphers. Once the
from plaintext to this ciphertext key was uncovered, the two individual
requires a cipher and a key. A wheels could be set accordingly.
316 CRYPTOGRAPHY
large amounts of information. on Thursday”, where “buy” is code
Longer messages were vulnerable for “kill” and “lemons” is code for
to another decryption technique a particular target on a hit list –
called frequency analysis. perhaps with all targets encoded
Initially developed by the Arab as fruits. Without the list of code
mathematician al-Kindi in the 9th words, deciphering the message’s Computer technology is
century, this technique made use full meaning is impossible. on the verge of providing
of the frequency of each letter of the the ability for [people] to
alphabet in a particular language. The Enigma code communicate and interact
The most common letter in the Another method of increasing the with each other in a totally
English language is “e”, so a security of encryption is to use a anonymous manner.
cryptanalyst would find the most polyalphabetic cipher, where a letter Peter Ludlow
common letter in the ciphertext in plaintext can be substituted for American philosopher
and designate that as e. The next several different letters in ciphertext,
most common letter is “t”, then “a”, thus removing the possibility of
and so on. Common groupings of frequency analysis. Such ciphers
letters, such as “th” and “ion” could were first developed in the 16th
also provide a way into revealing century, but the most famous one
the cipher. Given a large enough was the encryption produced by
ciphertext, this system worked on the Enigma machines used by the 10 plugs being connected correctly
any substitution cipher, no matter Axis forces in World War II. on the board. The settings became
how elaborate the encryption. The Enigma machine was a the encryption key. A three-rotor
There are two ways of formidable encryption device. In Enigma had over 158,962,555,217
combatting frequency analysis. essence, it was a battery connected billion possible settings, which
The first is to obscure the plaintext to 26 lightbulbs, or lamps – one for were changed daily.
by using a “code”. Cryptography each letter of the alphabet. When Enigma’s flaw was that it could
uses a specific definition of this a signaller pressed a letter on the not encrypt a letter as itself. This
term. A code changes an entire keyboard, a corresponding letter lit allowed Allied codebreakers to try
word or phrase in the plaintext up on the lampboard. Pressing the frequently used phrases, such as
before it is encrypted. An encoded same key a second time always lit “Heil Hitler” and “Weather Report”
plaintext might read “buy lemons a different lamp (never the same to attempt to figure out that day’s
letter as the key) because the key. Ciphertext without any of
connections between battery and the letters in those words was a
lampboard were altered by three potential ciphertext of them. Allied
rotors that clicked around with codebreakers used the Turing
every key press. Added complexity Bombe, an electromechanical
was introduced by the plugboard, device that mimicked Enigma
which swapped 10 pairs of letters, machines to break the encryption
thus scrambling the message by brute force, using shortcuts
further. To encrypt and decrypt an developed by British mathematician
Enigma message, both machines Alan Turing and others. The British
needed to be set up in the correct encryption device, the Typex, was
way. This involved the correct three a modified version of Enigma that
rotors being inserted and set to the could encode a letter as itself. The
right starting positions, and the Nazis gave up trying to crack it.

Asymmetric encryption
The Enigma machine was used
in German espionage between 1923 With symmetric encryption,
and 1945. The three rotor wheels sit messages are only as secure as
behind the lampboard, and the the key. This must be exchanged
plugboard is at the front. by physical means – written in a
 MODERN MATHEMATICS 317
modn), meaning the answer is
Public key Private key
just the remainder. So, for example,
a
if n were 10 and M were 12, that
would give the answer 2. If
M a were 2, it would also give an
answer of 2, because 10 goes into
2 zero times with a remainder of 2.
a
The answer to M modn is the
ciphertext (C), and in this example
abc def abc def it is 2. Someone spying could know
ghij klm 1a370255dezq3634xkylyh ghij klm the public key, n and a, but would
Encryption Decryption have no idea whether M is 2, 12,
or 1,002 (all divisible by 10 with
Public key encryption scrambles data with an encryption key a remainder of 2). Only Alice can
available to anyone. The data can only be unscrambled with a find out using her private key, z,
private key, known only to its owner. This method is effective for z
because C modn = M.
small amounts of data, but is too time-consuming for large amounts.
The crucial number in this
algorithm is n, which is formed by
military code book or whispered where a sender and receiver use multiplying two prime numbers: p
in the ear of a spy at a secluded two keys: one private and the other and q. Then a and z are calculated
rendezvous. If a key falls into the public. If two people, Alice and from p and q using a formula which
wrong hands, the encryption fails. Bob, wish to communicate in ensures that the modulo calculations
The rise of computer networks secret, Alice can send Bob her work. The only way to crack the
has allowed people to communicate public key. It is made up of two code is to figure out what p and q
easily over great distances without numbers, n and a. She keeps a are and then calculate z. To do that,
ever meeting. However, the most private key, z, to herself. Bob uses a codebreaker must figure out the
commonly used network, the n and a to encrypt a plaintext prime factors of n, but today’s RSA
internet, is public, so any symmetric message ( M), which is a string of algorithms use values for n with
key shared over a connection would numbers (or letters ciphered into 600 digits or more. It would take a
be available to unintended parties, numbers). Each plaintext number is supercomputer thousands of years
making it useless. The RSA raised to the power of a, and then to work out p and q by trial and
algorithm was an early development divided by n. The division is a error, making RSA and similar
in building asymmetric encryption, modulo operation (abbreviated to protocols practically unbreakable. ■

Finding primes in random ways

The RSA algorithm and other is raised to the power of p, and
public key encryption systems n is subtracted from the result,
require a large collection of primes the answer is a multiple of p.
to act as p and q. If the system Computers are not easily
relies heavily on too few primes, programmed to create truly
then it is possible for attackers to random sequences of numbers,
figure out some of the values for so companies use physical
p and q being used in everyday phenomena to generate them.
encryption. The solution is to have Computers are programmed
a supply of fresh primes. These to follow the movements of lava
are found by generating random lamps, measure radioactive
Lava lamps can be hooked up to numbers and testing their decay, or listen to white noise
computers in order to generate a primality with Pierre de Fermat’s made by radio transmissions,
selection of random numbers based “little theorem”: if a number (p) is turning that input into random
on their movements. prime, when another number (n) numbers to use for encryption.
318

JEWELS STRUNG
ON AN AS-YET
INVISIBLE
FINITE SIMPLE GROUPS
THREAD

S
imple groups have been multiplication, subtraction, or
IN CONTEXT described as algebra’s addition. In the early 1960s,
atoms. The Jordan-Hölder American mathematician Daniel
KEY FIGURE
theorem, proven around 1889, Gorenstein began to pioneer the
Daniel Gorenstein
asserts that, just as all positive classification of groups and issued
(1923–92)
integers can be constructed from his complete classification of finite
FIELD prime numbers, so all finite groups simple groups in 1979.
Number theory can be built from finite simple There are similarities between
groups. In mathematics, a group simple groups and symmetry in
BEFORE is not simply a collection of things, geometry. Just as a cube rotated
1832 Évariste Galois defines but a specification of how the group through 90 degrees looks the same
the concept of a simple group. members can be used to generate as it did before it was rotated, the
more members, for example, by transformations (rotational and
1869–89 Camille Jordan, a
French mathematician, and
Otto Hölder, a German, prove
that all finite groups can be A group is a set of elements (numbers, letters, or shapes)
built from finite simple groups. that are combined with other elements of the same group through
an operation (for example, addition, subtraction,
1976 Croatian mathematician or multiplication).
Swonimir Janko introduces the
sporadic simple group Janko
Group 4, the last finite simple
group to be discovered.
A group is finite if A group is simple if it
AFTER it has a finite number cannot be broken down
2004 American of elements. into smaller groups.
mathematicians Michael
Aschbacher and Stephen
D. Smith complete the
classification of finite
simple groups begun by Finite simple groups are the fundamental
Daniel Gorenstein. building blocks for all finite groups.
 MODERN MATHEMATICS 319
See also: The Platonic solids 48–49 ■ Algebra 92–99 ■ Projective geometry 154–55 ■ Group theory 230–33
■ Cryptography 314–17 ■ Proving Fermat’s last theorem 320–23

reflexive) associated with a regular This Cayley graph shows all 60 elements (different orientations) of the
2-D or 3-D shape can be arranged group A5 (the group of rotational symmetries of a regular icosahedron,
into a type of simple group known a three-dimensional shape with 20 faces), and how they relate to each
other. Since A5 has a finite number of elements, it is a finite group.
as a symmetry group. A5 is also a simple group. It has two generators (elements that can
be combined to give any other element of the group).
Infinite and finite groups
Some groups are infinite, as in the
group of all integers under addition, Generator g Generator h
which is infinite because numbers (blue arrow) (pink arrow)
can be added infinitely. However,
the numbers –1, 0, and 1 with the
multiplication operation form a
finite group; multiplying any
members of the group produces
only –1, 0, or 1. All the members of
a group and the rules for generating
it can be visualized using a Cayley
graph (see right).
A group is simple if it cannot be
broken down into smaller groups.
While the number of simple groups
is infinite, the number of types of
simple group is not – at least, not
when simple groups of finite size
are considered. In 1963, American
mathematician John G. Thompson Combining Combining element
proved that, with the exception of element with one
a a with one of
of the generating the generating
trivial groups (for example, 0 + 0 = 0, elements gives us
g elements gives us
h
or 1  1 = 1), all simple groups have this element . ga An element a this element ha.
an even number of elements. This
led Daniel Gorenstein to propose a
more difficult task: the classification Daniel Gorenstein In 1960–61, Gorenstein attended
of every finite simple group. a nine-month programme in
Born in Boston, Massachusetts, group theory at the University of
The Monster in 1923, Daniel Gorenstein had Chicago, which inspired him to
There are precise descriptions of taught himself calculus by the propose a classification of finite
18 families of finite simple groups, age of 12 and later attended simple groups. He continued to
with each family related to Harvard University. There, work on this project until his
symmetries of certain types of he became acquainted with death in 1992.
geometrical structure. There are finite groups, which would
also 26 individual groups called become his life’s work. After Key works
sporadic groups, the largest graduating in 1943, he stayed
at Harvard for several years, 1968 Finite groups
of which is called the Monster,
first to teach mathematics to 1979 “The classification
which has 196,883 dimensions and
military personnel during of finite simple groups”
approximately 8  1053 elements. World War II, then to earn his 1982 Finite simple groups
Every finite simple group either PhD under mathematician 1986 “Classifying the finite
belongs to one of the 18 families or Oscar Zariski. simple groups”
is one of the 26 sporadic groups. ■
320
IN CONTEXT

A TRULY
KEY FIGURE
Andrew Wiles (1953–)
FIELD

MARVELLOUS
Number theory
BEFORE
1637 Pierre de Fermat
states that there are no sets

PROOF
of positive whole numbers x, y,
and z that satisfy the equation
xn + yn = zn, where n is greater
than 2. However, he does not
provide the proof.

PROVING FERMAT’S LAST THEOREM 1770 Swiss mathematician

Leonhard Euler shows that
Fermat’s last theorem is true
when n = 3.
1955 In Japan, Yutaka
Taniyama and Goro Shimura
propose that every elliptic
curve has a modular form.
AFTER
2001 The Taniyama–Shimura
conjecture is established. It
becomes known as the
modularity theorem.

W
hen he died in 1665,
French mathematician
Pierre de Fermat left
behind a well-thumbed copy of
Arithmetica by the 3rd-century ce
Greek mathematician Diophantus,
its margins marked with Fermat’s
ideas. All the questions posed in
Fermat’s marginal scribbles were
later solved, except for one. He left
a tantalizing note in the margin: “I
have discovered a truly marvellous
proof, which this margin is too
small to contain here.”
Fermat’s note related to
Diophantus’s discussion of
Pythagoras’s theorem – that in a
right-angled triangle the square
 MODERN MATHEMATICS 321
See also: Pythagoras 36–43 ■ Diophantine equations 80–81 ■ Probability 162–65 ■ Elliptic functions 226–27 ■ Catalan’s
conjecture 236–37 ■ 23 problems for the 20th century 266–67 ■ Finite simple groups 318–19

for his doctoral thesis – a subject

Pierre de Fermat wrote a He claimed that that seemed to have little to do
note about Pythagoras’s xn + yn  zn for any with his interest in Fermat. Yet it
theorem in the margin positive integer n that was this branch of mathematics
of a book. is greater than 2. that would later enable Wiles to
prove Fermat’s last theorem.
In the mid-1950s, Japanese
mathematicians Yutaka Taniyama
and Goro Shimura had made the
“I have discovered a truly marvellous bold step of linking two apparently
proof, which this margin is too unrelated branches of mathematics.
small to contain here.” They claimed that every elliptic
curve (an algebraic structure) could
be associated with a unique
modular form, one of a class of
highly symmetrical structures
For more than three centuries, mathematicians tried belonging to number theory.
but failed to prove Fermat’s last theorem. The potential importance of
It was only solved in 1994. their conjecture was gradually
understood over the next three
decades and it became part of
an ongoing programme to link
of the hypotenuse (the side opposite Many mathematicians attempted different mathematical disciplines.
the right angle) is equal to the sum to reconstruct Fermat’s claimed However, no one had any idea how
of the squares on the other two proof after his death, or to find to prove it.
sides, or x2 + y2 = z2. Fermat knew their own. But despite the seeming In 1985, German mathematician
that this equation had an infinity simplicity of the problem, no one was Gerhard Frey made a link between
of integer solutions for x, y, and z, successful, although a century later the conjecture and Fermat’s ❯❯
such as 3, 4, and 5 (9 + 16 = 25) Leonhard Euler did prove the theory
and 5, 12, and 13 (25 + 144 = 169), where n = 3.
known as “Pythagorean triples”. y
He then wondered if other triples Finding a solution
could be found to the power of 3, Fermat’s last theorem remained one
4, or any integer beyond 2. The of the great unsolved problems in
conclusion Fermat reached was mathematics for more than 300
that no integer greater than 2 could years, until it was proved by British x
stand for n. Fermat wrote: “It is mathematician Andrew Wiles in
impossible for a cube to be the 1994. Wiles had first read about
sum of two cubes, a fourth power Fermat’s challenge when he was
[number to the power of 4] to be ten. He had been amazed that he,
the sum of two fourth powers, or just a boy, could make sense of it,
in general for any number that is a and yet the best mathematical
power greater than the second to minds in the world had failed to
be the sum of two like powers.” prove it. It fuelled his motivation to
Wiles’s investigation of Fermat’s
Fermat never revealed the proof he read mathematics at the University last theorem began with his study of
claimed to have for his theory and of Oxford, and then to take his PhD elliptic curves, which are described by
so it remained unsolved, becoming at Cambridge. There, he chose the equation y2 = x3 + Ax + B, where
known as Fermat’s last theorem. elliptic curves as the area of study A and B are constants (fixed).
322 PROVING FERMAT’S LAST THEOREM
Wiles revealed that he had been
Gerhard Frey working on Fermat while still
proposed thatn any n carrying out his daily tasks of
n The Taniyama–Shimura lecturing, writing, and teaching.
case of x + y = z
conjecture stated that every Wiles recruited the help of these
(where n > 2) would lead
elliptic curve is modular. colleagues for the final step in
to an elliptic curve that
is not modular. compiling his proof. He turned to
American mathematician Nick Katz
to check his reasonings. Katz could
find no errors, so Wiles decided to go
public. In June 1993, at a conference
Andrew Wiles proved that every elliptic curve at the University of Cambridge,
is modular. Wiles delivered his results. Tension
rose as he piled his results one on
top of the other, with only one end
in view. He delivered his final line,
“Which proves Fermat’s last
theorem”, smiled, and added,
n n n
This proved that there are no cases of x + y = z “I think I’ll leave it there.”
(where n > 2).
Fixing an error
The next day, the world’s press
was full of the story, transforming
Wiles into the world’s most famous
mathematician. Everyone wanted
Wiles proved Fermat’s last theorem. to know how this problem had
finally been solved. Wiles was
delighted, but then came a twist;
there was a problem with his proof.
The results had to be verified
last theorem. Working from a seemingly impossible Taniyama– before they could be published –
hypothetical solution to the Fermat Shimura conjecture, then he would and Wiles’s proof covered scores of
equation, he constructed a curious also prove Fermat’s last theorem. pages. Among the reviewers was
elliptic curve that appeared not to Unlike most mathematicians, who
be modular. He argued that such like to work collaboratively, Wiles
a curve could only exist if the decided to pursue this goal on his
Taniyama–Shimura conjecture own, telling no one except his wife.
were false, in which case Fermat’s He felt that to talk openly about
last theorem would also be working on Fermat would stir up
false. On the other hand, if the excitement in the mathematics Some mathematics problems
Taniyama–Shimura conjecture community, and perhaps lead to look simple. There’s no reason
were true, Fermat’s last theorem unwanted competition. However, why these problems
would follow. In 1986, Ken Ribet, a as the proof reached its final stages, shouldn’t be easy,
professor at Princeton University, in the seventh year of working on it, and yet they turn out
in New Jersey, managed to prove Wiles realized he needed help. to be extremely intricate.
Frey’s conjectured link. At the time, Wiles was Andrew Wiles
employed at the Institute for
Proving the unprovable Advanced Study (IAS) in Princeton,
Ribet’s proof electrified Wiles. home to some of the world’s finest
Here was the chance he had been mathematicians. These colleagues
waiting for – if he could prove the were completely astounded when
 MODERN MATHEMATICS 323
alone, the error might have been Katz and the wider mathematical
identified earlier. The world community were now convinced
believed that Wiles had resolved there were no mistakes, and
Fermat’s last theorem, and it was Wiles emerged for a second time
waiting for the finished, published as the conqueror of Fermat’s last
I had this rare privilege proof. Wiles was under immense theorem – this time on solid ground.
of being able to pursue pressure. His mathematical
in my adult life achievements so far had been After the theorem
what had been impressive, but his reputation was Fermat was amazingly far-sighted
my childhood dream. at stake. Day after day, Wiles tried in his original conjecture, but it is
Andrew Wiles different approaches to the problem, unlikely that the “marvellous proof”
which proved futile – as his fellow he claimed to have discovered
IAS mathematician Peter Sarnak existed. The idea that every
said, “It was like pinning down a mathematician since the 17th
carpet in one corner of a room, only century could have missed
for the carpet to pop up in another.” a proof that a mathematician from
Eventually, Wiles turned to a friend, Fermat’s time could have discovered
Wiles’s friend Nick Katz. For a British algebra specialist Richard is inconceivable. In addition, Wiles
whole summer Katz went through Taylor, and the pair worked solved the theorem using advanced
the proof line by line, querying and together on the proof for the next mathematical tools and ideas
questioning until the meaning was nine months. invented long after Fermat.
clear. One day, he thought he had Wiles was close to having to In many ways, it is not the
spotted a hole in the argument. He admit that he had claimed a proof proving of Fermat’s last theorem
emailed Wiles, who replied, but not prematurely. Then, in September that has significance, but rather
to Katz’s satisfaction. More emails 1994, he had a revelation. If he took the proofs used by Wiles. A
followed, before the truth emerged – his present problem-solving method seemingly impossible problem
Katz had found a flaw at the heart and added its strengths to an earlier about integers had been solved
of Wiles’s work. A vital point in approach of his, then one might fix by marrying number theory to
the proof contained an error that the other, allowing him to solve the algebraic geometry, using new and
undermined Wiles’s method. problem. It seemed a small insight, existing techniques. This in turn
Suddenly Wiles’s approach but it made all the difference. opened up new ways of looking
was brought into question. Had Within weeks, Wiles and Taylor had at how to prove many other
he worked with others rather than plugged the gap in the proof. Nick mathematical conjectures. ■

Andrew Wiles The son of an Anglican priest including the Taniyama–

who later became a professor Shimura conjecture. He also
of divinity, Wiles was born in began his long solo attempt to
Cambridge in 1953, and was prove Fermat’s last theorem.
a passionate problem-solver in His eventual success led to
mathematics from an early age. him receiving the Abel Prize –
Awarded his first degree in the highest honour in
mathematics at Merton College, mathematics – in 2016.
Oxford, and his doctorate at Clare Wiles has also taught in
College, Cambridge, he took up a Bonn and Paris, and at the
post at the Institute for Advanced University of Oxford, where
Study in Princeton in 1981, and he was appointed Regius
was appointed professor there Professor of Mathematics in
the following year. 2018. A new mathematics
While in the US, Wiles made building at Oxford – as well
contributions to some of the most as an asteroid – 9999 Wiles –
elusive problems in his field, have been named after him.
324

NO OTHER
RECOGNITION
IS NEEDED
PROVING THE POINCARÉ CONJECTURE

I
n 2000, the Clay Mathematics
IN CONTEXT Institute in the US celebrated
The “3-sphere” is a 3-D
spherical surface existing the millennium with seven
KEY FIGURE
within 4 dimensions. prize problems. Among them was
Grigori Perelman (1966–)
the Poincaré conjecture, which
FIELDS had challenged mathematicians
Geometry, topology for nearly a century. Within a few
years, it was solved – by a little-
BEFORE known Russian mathematician,
1904 Henri Poincaré states his Grigori Perelman.
conjecture on the equivalence Poincaré argued that Poincaré’s conjecture, conceived
of shapes in 4-D space. any 3-D shape without by the French mathematician in
holes can be distorted
1904, is stated as: “Every simply
1934 British mathematician to form the 3-sphere.
connected, closed 3-manifold is
Henry Whitehead stirs interest homeomorphic to the 3-sphere”.
in Poincaré’s conjecture by In topology, a field that studies the
publishing an erroneous proof. geometrical properties, structure,
1960 American mathematician and spatial relations of shapes, a
Stephen Smale proves the sphere (a 3-D object in geometry)
conjecture is true in the fifth is said to be a 2-manifold with a
His conjecture can be
and higher dimensions. 2-D surface existing within a 3-D
extended to any number
of dimensions. space – a solid ball, for example. A
1982 Poincaré’s conjecture is 3-manifold, such as the 3-sphere,
proved in four dimensions by is a purely theoretical concept: it
American mathematician has a 3-D surface and exists in a
Michael Freedman. 4-D space. The description “simply
connected” means that the figure
AFTER has no holes, unlike a bagel or hoop
2010 When Perelman rejects Perelman’s
shape (torus), and “closed” means
the Clay Millennium Prize, the proof of the
the shape is limited by boundaries,
£1 million award is used to set Poincaré conjecture unlike the open endlessness of an
up the Poincaré Chair for gifted was confirmed infinite plane. In topology, two
young mathematicians. in 2006. figures are homeomorphic if they
can be distorted or stretched into
 MODERN MATHEMATICS 325
See also: The Platonic solids 48–49 ■ Graph theory 194–95 ■ Topology 256–59
■ Minkowski space 274–75 ■ Fractals 306–11

A 3-sphere is the 3-D equivalent

of a spherical surface, that is a
two-dimensional surface, or 2-sphere,
such as the ball shown here. To
appreciate the shape of the ball, it
has to be viewed in 3-D space. To
see a 3-sphere requires 4-D space.

technique called surgery, in effect

cutting out the singularities, and
was able to prove the conjecture. Grigori Perelman
Surprising the maths world Born in 1966 in St Petersburg,
Perelman had achieved success Grigori Perelman developed a
the same shape. While the question quietly. Unconventionally, he passion for mathematics from
of whether every closed 3-manifold posted his first 39-page paper on his mother, who taught the
could be deformed to take the the subject online in 2002, emailing subject. Aged 16, he won a
shape of a 3-sphere is hypothetical, a summary to 12 mathematicians gold medal at the International
Perelman has claimed that it holds in the US. He published two more Mathematical Olympiad in
the key to understanding the shape instalments a year later. Others Budapest, achieving a perfect
of the Universe. reconstructed his results and score. A successful academic
explained them in the Asian career followed, including
a spell at several research
Finding a solid proof Journal of Mathematics. Finally,
institutes in the US, where
Initially, it proved easier to his proof was fully accepted by the
he solved a major geometry
substantiate the conjecture for mathematical community in 2006. problem called the Soul
manifolds of the fourth, fifth, and Since then, Perelman’s work has conjecture. While there, he
higher dimensions than it was for been closely studied, fuelling new met Richard Hamilton, whose
3-manifolds. In 1982, American developments in topology, including work influenced his proof of
mathematician Richard Hamilton a more powerful version of his and the Poincaré conjecture.
attempted to prove the conjecture Hamilton’s technique for using The reclusive Perelman did
using Ricci flow, a mathematical Ricci flow to smooth singularities. ■ not enjoy the fame his proof
process that potentially allows any brought him. He turned down
4-D shape to be distorted to an the two greatest accolades for
increasingly smooth version, and a mathematician: the Fields
ultimately to a 3-sphere. However, Medal in 2006 and the Clay
the flow failed to handle spike-like Mathematics Institute prize
“singularities” – deformities (and its $1 million award) in
Perelman’s proof… solved a 2010, saying it belonged as
including “cigars” and infinitely
problem that for more than a much to Hamilton.
dense “necks”.
Perelman, who learned much century was an indigestible
from Hamilton during a two-year seed at the core of topology. Key works
fellowship at Berkeley in the early Dana Mackenzie
American science writer 2002 “The entropy formula
1990s, continued to study Ricci
for the Ricci flow and its
flow and its application to the
geometric applications”
Poincaré conjecture when he 2003 “Finite extinction time for
returned to Russia. He masterfully the solutions to the Ricci flow
overcame the limitations that on certain 3-manifolds”
Hamilton encountered by using a
DIRECTO
RY
328

DIRECTORY
I
n addition to the mathematicians covered in the preceding chapters
of this book, many other men and women have made an impact on
the development of mathematics. From the ancient Egyptians,
Babylonians, and Greeks to the medieval scholars of Persia, India, and
China and the city-state rulers of Renaissance Europe, those looking to
build, trade, fight wars, and manipulate money realized that measuring
and calculating were crucial. By the 19th and 20th centuries, mathematics
had become a global discipline, with its practitioners involved in all the
sciences. Maths remains crucial in the 21st century as space exploration,
medical innovations, artificial intelligence, and the digital revolution press
ahead, and more secrets about the Universe are revealed.

The Lune of Hippocrates, as it was later descriptions of a steam-powered device

THALES OF MILETUS called, is bounded by the arcs of two called an aeolipile, a wind wheel that
c. 624–c. 545 bce circles, the smaller of which has as its could operate an organ, and a vending
diameter a chord spanning a right angle machine that dispensed “holy” water.
Thales lived in Miletus, an ancient on the larger circle. His mathematical accomplishments
Greek city in what is now Turkey. A See also: Pythagoras 36–43 ■ Euclid’s included describing a method for
student of mathematics and astronomy, Elements 52–57 ■ Trigonometry 70–75 computing the square roots and cubic
he broke with the tradition of using roots of numbers. He also devised a
mythology as a way of explaining the formula for finding the area of a triangle
world. Thales used geometry to EUDOXUS OF CNIDUS from the lengths of its sides.
calculate the height of pyramids and c. 390–c. 337 bce See also: Euclid’s Elements 52–57
the distance of ships from the shore. ■ Trigonometry 70–75 ■ Cubic

The theorem named after him states Eudoxus lived in the Greek city of equations 102–05
that where the longest side of a triangle Cnidus (now in Turkey). He developed
contained within a circle is the diameter the “method of exhaustion” to prove
of the circle, that triangle has to be a statements about areas and volumes by ARYABHATA
right-angled triangle. The astronomical successive approximations. For example, 476–550 ce
discoveries attributed to Thales include he was able to show that the areas of
his forecast of the 585 bce solar eclipse. circles relate to each other according A Hindu mathematician and astronomer,
See also: Pythagoras 36–43 ■ Euclid’s to the squares of their radii; that the Aryabhata worked in Kusumapara, an
Elements 52–57 ■ Trigonometry 70–75 volumes of spheres relate to each other Indian centre of learning. His verse
according to the cubes of their radii; treatise Aryabhatiya contains sections
and that the volume of a cone is one-third on algebra and trigonometry, including
HIPPOCRATES OF CHIOS that of a cylinder of the same height. an approximation for pi () of 3.1416,
c. 470–c. 410 bce See also: The Rhind papyrus 32–33 accurate to four decimal places.
■ Euclid’s Elements 52–57 ■ Calculating Aryabhata also correctly believed pi
Originally a merchant on the Greek pi 60–65 to be irrational. He calculated Earth’s
island of Chios, Hippocrates later moved circumference as a distance close to the
to Athens, where he first studied, then current accepted figure. He also defined
practised mathematics. References by HERO OF ALEXANDRIA some trigonometric functions, produced
later scholars suggest that he was c. 10–c. 75 ce complete and accurate sine and cosine
responsible for the first systematic tables, and calculated solutions to
compilation of geometrical knowledge. A native of Alexandria in the Roman simultaneous quadratic equations.
He was able to calculate the area of province of Egypt, Hero (or Heron) See also: Quadratic equations 28–31
crescent-shaped figures contained was an engineer, inventor, and ■ Calculating pi 60–65 ■ Trigonometry

within intersecting circles (lunes). mathematician. He published 70–75 ■ Algebra 92–99

 DIRECTORY 329

equations of the second order (to the poor backgrounds were subsidized by
BHASKARA I power of two), which would not be the royal estate. He later became dean
c. 600–c. 680 solved in Europe until the 18th century. of Rouen Cathedral. Oresme devised
See also: Quadratic equations 28–31 a coordinate system with two axes to
Little is known about Bhaskara I, ■ Diophantine equations 80–81 ■ Cubic represent the change of one quality with
although he may have been born in the equations 102–05 respect to another – for example, how
Saurastra region on India’s west coast. temperature changes with distance.
He became one of the most important He worked on fractional exponents and
scholars of the astronomy school founded NASIR AL-DIN AL-TUSI infinite series and was the first to prove
by Aryabhata (see page 328), and wrote 1201–74 the divergence of harmonic series, but
a commentary, Aryabhatiyabhasya, on his proof was lost and the theory was
Aryabhata’s earlier Aryabhatiya treatise. Born in Tus, the Persian mathematician not proven again until the 17th century.
Bhaskara I was the first person to write al-Tusi devoted his life to study after He also argued that Earth could be
numbers in the Hindu-Arabic decimal he lost his father at a young age. He rotating in space, rather than the
system with a circle for zero. In 629, he became one of the great scholars of his Church-approved view that the
also found a remarkably accurate day, making important discoveries in celestial bodies circled around Earth.
approximation of the sine function. maths and astronomy. He established See also: Algebra 92–99 ■ Coordinates
See also: Trigonometry 70–75 trigonometry as a discipline, and in his 144–151 ■ Calculus 168–75
■ Zero 88–91 Commentary on the Almagest – an
introduction to trigonometry – described
methods for calculating sine tables. NICCOLÒ FONTANA TARTAGLIA
IBN AL-HAYTHAM Although taken prisoner by invading 1499–1557
c. 965–c. 1040 Mongols in 1255, al-Tusi was appointed
a scientific advisor by his captors and As a child, Tartaglia was attacked by
Also known as Alhazen, Ibn al-Haytham later established an astronomical French soldiers invading Venice. He
was an Arab mathematician and observatory in the Mongol capital survived, but with serious facial injuries
astronomer, born in Basra, now in Iraq, Maragheh, now in Iran. and a speech impediment, which earned
who worked at the court of the Fatimid See also: Trigonometry 70–75 him the nickname “Tartaglia”, or
Caliphate in Cairo. He was a pioneer of stammerer. Essentially self-taught,
the scientific method that maintained he became a civil engineer, designing
that hypotheses should be tested by KAMAL AL-DIN AL-FARISI fortifications. Tartaglia realized that
experiment and not just assumed to c. 1260–c. 1320 an understanding of the trajectory of
be true. Among his achievements, he cannonballs was critical for his designs,
established the beginnings of a link Al-Farisi was born in Tabriz, Persia which led him to pioneer the study of
between algebra and geometry, building (now Iran). He was a student of polymath ballistics. His published mathematical
on the work of Euclid and trying to Qutb al-Din al-Shirazi, himself a pupil works included a formula for solving
complete the lost eighth volume of of Nasir al-Din al-Tusi (see above), cubic equations, an encyclopaedic
Apollonius of Perga’s Conics. and, like them, was a member of the maths treatment – Treatise on Numbers
See also: Euclid’s Elements 52–57 Maragheh school of mathematician– and Measures – and translations of
■ Conic sections 68–69 astronomers. His explorations of number Euclid and Archimedes.
theory included amicable numbers and See also: The Platonic solids 48–49
factorization. He also applied the theory ■ Trigonometry 70–75 ■ Cubic

BHASKARA II of conic sections (circles, ellipses, equations 102–05 ■ The complex plane
1114–85 parabolas, and hyperbolas) to solve 214–15
optical problems, and explained that
One the greatest of the medieval Indian the different colours of a rainbow were
mathematicians, Bhaskara II was born produced by the refraction of light. GEROLAMO CARDANO
in Vijayapura, Karnataka, and is believed See also: Conic sections 68–69 1501–76
to have become the head of the ■ The binomial theorem 100–01

astronomical observatory at Ujjain in A contemporary of Niccolò Tartaglia,

Madhya Pradesh. He introduced some Cardano was born in Lombardy and
preliminary concepts of calculus; NICOLE ORESME became an outstanding physician,
established that dividing by zero yields c. 1320–82 astronomer, and biologist, as well as
infinity; found solutions to quadratic, a renowned mathematician. He studied
cubic, and quartic equations (including Born in Normandy, France, probably to at the universities of Pavia and Padua
negative and irrational solutions); and a peasant family, Oresme studied at the in what is now Italy, was awarded a
suggested ways to unlock Diophantine College of Navarre, where pupils from doctorate in medicine, and worked as
330 DIRECTORY

a physician before becoming a teacher he would not share them with other
of mathematics. Cardano published a mathematicians. Many of these ideas JOHANN LAMBERT
solution to cubic and quartic equations, were published by l’Hôpital in 1728–77
acknowledged the existence of Infinitesimal Calculus.
imaginary numbers (based on the See also: Calculus 168–75 Lambert was a Swiss-German
square root of –1), and is alleged to have polymath, born in Mulhouse (now in
forecast the exact date of his own death. France), who taught himself maths,
See also: Algebra 92–99 ■ Cubic JEAN LE ROND D’ALEMBERT philosophy, and Asian languages. He
equations 102–05 ■ Imaginary and 1717–83 worked as a tutor before becoming a
complex numbers 128–31 member of the Munich Academy in 1759
The illegitimate son of a celebrated Paris and the Berlin Academy five years later.
hostess, d’Alembert was brought up by Among his mathematical achievements,
JOHN WALLIS a glazier’s wife. Funded by his estranged he provided rigorous proof that pi is an
1616–1703 father, he studied law and medicine, irrational number, and introduced
then turned to mathematics. In 1743, he hyperbolic functions into trigonometry.
Although Wallis studied medicine at stated that Newton’s third law of motion He produced theorems on conic
Cambridge University and was later is as true for freely moving bodies as it is sections, simplified the calculation of
ordained a priest, he retained the for fixed bodies (d’Alembert’s principle). the orbits of comets, and created several
interest in arithmetic he first developed He also developed partial differential new map projections. Lambert also
as a schoolboy in Kent. A supporter of equations, explained the variations in invented the first practical hygrometer,
the Parliamentarian cause, Wallis the orbits of Earth and other planets, and used to measure the humidity of air.
deciphered Royalist dispatches during researched integral calculus. Like other See also: Calculating pi 60–65 ■ Conic
the English Civil War. In 1644, he was French philosophes, such as Voltaire and sections 68–69 ■ Trigonometry 70–75
appointed professor of geometry at the Jean-Jacques Rousseau, d’Alembert
University of Oxford and became a believed in the supremacy of human
champion of arithmetic algebra. His reason over religion. GASPARD MONGE
contributions towards the development See also: Calculus 168–75 ■ Newton’s 1746–1818
of calculus include originating the idea laws of motion 182–83 ■ The algebraic
of the number line, introducing the resolution of equations 200–01 The son of a merchant, by the age of 17
symbol for infinity, and developing Monge was teaching physics in Lyon,
standard notation for powers. He was France. He later worked as a draftsman
one of the small group of scholars whose MARIA GAETANA AGNESI at the École Royale, Mézières, and in
meetings led to the establishment of 1718–99 1780 became a member of the Academy
the Royal Society of London in 1662. of Sciences. Monge was active in public
See also: Conic sections 68–69 Born in Milan, then under Austrian life, embracing the ideals of the French
■ Algebra 92–99 ■ The binomial Hapsburg rule, Agnesi was a child Revolution. He was appointed Minister
theorem 100–01 ■ Calculus 168–75 prodigy who, as a teenager, lectured of the Marine in 1792, and also worked
friends of her father on a wide range to reform France’s education system,
of scientific subjects. In 1748, Agnesi helping to found the École Polytechnique
GUILLAUME DE L’HÔPITAL became the first woman to write a in Paris in 1794 and contributing to
1661–1704 maths textbook, the two-volume the founding of the metric system of
Instituzioni analitiche (Analytical measurement in 1795. Described as the
Born in Paris, l’Hôpital was interested in Institutions), which covered arithmetic, “father of engineering drawing”, Monge
maths from a young age and was elected algebra, trigonometry, and calculus. invented descriptive geometry, the
to the French Academy of Sciences in Two years later, recognizing her mathematical basis of technical
1693. Three years later, he published the achievement, Pope Benedict XIV drawing, and orthographic projection.
first textbook on infinitesimal calculus: awarded her the chair of mathematics See also: Decimals 132–37 ■ Projective
Analyse des infiniment petits pour and natural philosophy at the University geometry 154–55 ■ Pascal’s triangle
l’intelligence des lignes courbes of Bologna, making her the first woman 156–61
(Analysis of the Infinitesimally Small professor of maths at any university.
for the Understanding of Curved Lines). The equation describing a particular
Although l’Hôpital was an accomplished bell-shaped curve called the “witch ADRIEN-MARIE LEGENDRE
mathematician, many of his ideas were of Agnesi” is named in her honour, 1752–1833
not original. In 1694, he had offered the although “witch” was a mistranslation
Swiss mathematician Johann Bernoulli from the Italian word for “curve”. Legendre taught physics and maths at
300 livres a year for information on his See also: Trigonometry 70–75 the École Militaire in Paris from 1775 to
latest discoveries and an agreement that ■ Algebra 92–99 ■ Calculus 168–75 1780. During this period, he also worked
 DIRECTORY 331

on the Anglo-French Survey, using Christiana (now Oslo) in 1822, he becoming a professor of mathematics
trigonometry to calculate the distance travelled widely in Europe, visiting at the Humboldt University of Berlin.
between the Paris Observatory and leading mathematicians. He returned Weierstrass was a pioneer in the
London’s Royal Greenwich Observatory. to Norway in 1828, but died from development of mathematical analysis
During the French Revolution, he lost his tuberculosis the following year at the and in the modern theory of functions,
private fortune but in 1794 he published age of 26, days before a letter arrived and rigorously reformulated calculus.
Eléments de géométrie (Elements of offering him a prestigious maths An influential teacher, he included
Geometry), which remained a key professorship at the University of Berlin. among his pupils the young Russian
geometry textbook for the next century, Abel’s most important mathematics émigré and pioneering mathematician
and he was then appointed a maths contribution was to prove that there is Sofya Kovalevskaya (see page 332).
examiner at the École Polytechnique. no general algebraic formula for solving See also: Calculus 168–75 ■ The
In number theory, he conjectured the all quintic (fifth-degree) equations. fundamental theorem of algebra 204–09
quadratic reciprocity law and the prime To make his proof, he invented a type
number theorem. He also produced the of group theory where the order of the
least-squares method for estimating a elements within a group is immaterial. FLORENCE NIGHTINGALE
quantity based on consideration of This is now known as an abelian group. 1820–1910
measurement errors, and gave his name The annual Abel Prize for mathematics
to three forms of elliptic integrals – the is awarded in his honour. Named after her Italian birthplace,
Legendre transform, transformation, See also: The fundamental theorem Florence Nightingale was a British social
and polynomials. of algebra 204–09 ■ Elliptic functions reformer and pioneer of modern nursing,
See also: Calculus 168–75 ■ The 226–27 ■ Group theory 230–33 who based much of her work on the use
fundamental theorem of algebra 204–09 of statistics. In 1854, after the outbreak
■ Elliptic functions 226–27 of the Crimean War, Nightingale went
JOSEPH LIOUVILLE to work among wounded soldiers at
1809–82 The Barrack Hospital in Scutari, Turkey.
SOPHIE GERMAIN There, she campaigned tirelessly for
1776–1831 Born in northern France, Liouville better hygiene, earning the nickname
graduated from the École Polytechnique, “The lady with the lamp”. Back in
During the chaos of the French Paris, in 1827 and took up a teaching Britain, Nightingale became an
Revolution, 13-year-old Sophie Germain post there in 1838. His academic work innovator in the use of graphs to display
was confined to her wealthy father’s spanned number theory, differential statistical data. She developed the
house in Paris and began to study the geometry, mathematical physics, and Coxcomb chart, a variation on the pie
mathematics books in his library. As a astronomy, and in 1844 he was the first chart, using circle segments of different
woman she was ineligible to study at the to prove the existence of transcendental sizes to display variations in data, such
École Polytechnique, but she obtained numbers. Liouville wrote more than as the causes of mortality among
lecture notes and corresponded with the 400 papers and in 1836 founded the soldiers. Her actions helped to establish
mathematician Joseph-Louis Lagrange. Journal de Mathématiques Pures et a Royal Commission on health in the
In her work on number theory, Germain Appliquées (Journal of Pure and Applied army in 1856. In 1907, she was the first
also corresponded with Adrien-Marie Mathematics), the world’s second-oldest woman to receive the Order of Merit,
Legendre (see above) and Carl Gauss, mathematical journal, which is still Britain’s highest civilian honour.
and her ideas on Fermat’s last theorem published monthly. See also: The birth of modern statistics
helped Legendre to prove the theorem See also: Calculus 168–75 ■ The 268–71
where = 2. In 1816, she was the first
n fundamental theorem of algebra 204–09
woman to win a prize from the Academy ■ Non-Euclidean geometries 228–29

of Sciences in Paris, for a paper on the ARTHUR CAYLEY

elasticity of metal plates. 1821–95
See also: The fundamental theorem KARL WEIERSTRASS
of algebra 204–09 ■ Proving Fermat’s 1815–97 Born in Richmond, Surrey, Cayley
last theorem 320–23 was probably the leading British pure
Born in Westphalia, Germany, mathematician of the 19th century.
Weierstrass developed an interest in Graduating from Trinity College,
NIELS ABEL mathematics at an early age. His parents Cambridge, he embarked on a career
1802–29 wanted their son to have a career in as a conveyancing lawyer. In 1860,
administration, so he was sent to study however, he gave up his lucrative
Abel was a Norwegian mathematician law and economics at university but law practice to take up a pure maths
who died tragically young. After left without gaining a degree. He then professorship at Cambridge, on a far
graduating from the University of trained as a teacher, ultimately more modest salary. Cayley was a
332 DIRECTORY

pioneer of group theory and matrix

algebra, devised the theories of higher GOTTLOB FREGE GIUSEPPE PEANO
singularities and invariants, worked in 1848–1925 1858–1932
higher-dimensional geometry, and
extended the quaternions of William The son of the principal of a girls’ Brought up on a farm in the northern
Hamilton to create octonions. school in Wismar, northern Germany, Italian region of Piedmont, Peano
See also: Non-Euclidean geometries Frege studied mathematics, physics, studied at the University of Turin, where
228–29 ■ Group theory 230–33 chemistry, and philosophy at the he gained his doctorate in maths in
■ Quaternions 234–35 ■ Matrices 238–41 universities of Jena and Göttingen. 1880. Almost immediately, he began to
He then spent his whole working life teach infinitesimal calculus at the same
teaching mathematics in Jena. He institution, where he was appointed a
RICHARD DEDEKIND lectured in all areas of mathematics, full professor in 1889. Peano’s first
1831–1916 specializing in calculus, but wrote textbook, on calculus, was published in
mostly on the philosophy of the subject, 1884, and in 1891 he began work on the
Dedekind was one of Carl Gauss’s bringing the two disciplines together to five-volume Formulario Mathematico
students at the University of Göttingen, almost single-handedly invent modern (Formulation of Mathematics), which
Germany. After graduating, he worked mathematical logic. He once commented contained the fundamental theorems
as an unsalaried lecturer before teaching that “Every good mathematician is of maths in a symbolic language largely
at the Zurich Polytechnic, Switzerland. at least half a philosopher, and every developed by him. Many of the symbols
Returning to Germany, in 1862 he good philosopher at least half a and abbreviations are still in use today.
started work at the Technical High mathematician”. Frege mixed little He devised axioms for natural numbers
School in Braunschweig, where he with students or colleagues and (Peano axioms), developed natural logic
remained for the rest of his working life. remained largely unrecognized in and set theory notation, and contributed
He proposed the Dedekind cut, now his lifetime, although he was a major to the modern method of mathematical
a standard definition of real numbers, influence on the work of Bertrand induction, used as a proof technique.
and defined concepts of set theory, Russell, Ludwig Wittgenstein and See also: Calculus 168–75 ■ Non-
such as similar sets and infinite sets. other mathematical logicians. Euclidean geometries 228–29 ■ The
See also: The fundamental theorem of See also: The logic of mathematics logic of mathematics 272–73
algebra 204–09 ■ Group theory 230–33 272–73 ■ Fuzzy logic 300–01
■ Boolean algebra 242–47

NIELS VON KOCH

SOFYA KOVALEVSKAYA 1870–1924
MARY EVEREST BOOLE 1850–91
1832–1916 Born in Stockholm, Sweden, Koch
Moscow-born Kovalevskaya was the first studied at the universities of Stockholm
Mary Everest’s love of maths began woman in Europe to gain a doctorate in and Uppsala, later becoming professor
young when she studied the books mathematics, the first woman to join the of mathematics at Stockholm. He is
in the study of her clergyman father, editorial board of a scientific journal, best known for the fractal – Von Koch’s
whose friends included polymath and the first woman to be appointed “snowflake” curve – he described in a
Charles Babbage, the inventor of the a professor of maths. She achieved 1906 paper. This fractal is constructed
Difference Engine. At 18, Mary met all this despite being barred from a from an equilateral triangle in which the
renowned mathematician George Boole university education in her native Russia central third of each side is replaced by
(who, like her, was self-taught) in Ireland. because of her gender. Aged 17, Sofya the base of another equilateral triangle,
They married five years later, but George eloped with palaeontologist Vladimir with this process continuing indefinitely.
died soon after the birth of their fifth Kovalevsky to Germany, where she If all the triangles face outwards, the
child. In 1864, with five daughters to studied at the University of Heidelberg resulting curve takes on the appearance
raise and no financial support, Mary and then Berlin, where she received of a snowflake.
returned to London, where she worked tuition from German mathematician Karl See also: Fractals 306–11
as a librarian at Queen’s College, Weierstrass (see page 331). Her doctorate
a girls’ school, and later gained a was awarded for three papers, the most
reputation as an eminent children’s significant being on partial differential ALBERT EINSTEIN
teacher. She also wrote books that equations. Sofya ended her career as a 1879–1955
made maths more accessible to young professor of maths at the University of
students, including Philosophy and Stockholm, where she died of influenza Einstein was an outstandingly gifted
Fun of Algebra (1909). aged just 41. physicist and mathematician. Born in
See also: Algebra 92–99 ■ The See also: Calculus 168–75 ■ Newton’s Germany, he moved with his family
fundamental theorem of algebra 204–09 laws of motion 182–83 to Italy when young and studied in
 DIRECTORY 333

Switzerland. In 1905, he was awarded Her doctorate was awarded by The the other loses. The theory provided
his doctorate by the University of Zurich Catholic University of America in 1943 insights into complex systems in daily
and published ground-breaking papers for a dissertation on set theory. In 1959, life such as economics, computing, and
on Brownian motion, the photoelectric Lofton Haynes received a Papal medal the military. He also created a design
effect, special and general relativity, and for her contributions to education and model for modern computer architecture,
the equivalence of matter and energy. community activism, and in 1966 she and worked in quantum and nuclear
In 1921, he was awarded the Nobel Prize was the first woman to chair the District physics, contributing to the atomic
for his contribution to physics, and he of Columbia State Board of Education. bomb during World War II.
continued to develop the understanding See also: The logic of mathematics See also: The logic of mathematics
of quantum mechanics in the years 272–73 272–73 ■ The Turing machine 284–89
that followed. Because of his Jewish
background, he did not return to
Germany after Hitler came to power in MARY CARTWRIGHT GRACE HOPPER
1933, but settled in the United States, 1900–98 1906–92
becoming a citizen there in 1940.
See also: Newton’s laws of motion The daughter of an English country Born Grace Murray in New York City,
182–83 ■ Non-Euclidean geometries vicar, Cartwright was one of the first Hopper was a pioneering computer
228–29 ■ Topology 256–59 mathematicians to investigate what programmer. After gaining a doctorate
■ Minkowski space 274–75 would later be known as chaos theory. from Yale University in 1934, she taught
She graduated from the University of for several years before the outbreak of
Oxford in 1923 with a degree in World War II. When her application to
L. E. J. BROUWER mathematics. Seven years later, her enlist in the US Navy was rejected, she
1881–1966 doctoral thesis was examined by joined the Naval Reserve and began her
mathematician John E. Littlewood, with transition to computer science. After
Luitzen Egbertus Jan Brouwer (known whom she would have a long academic the war, while employed as a senior
to his friends as “Bertus”) was born in collaboration, especially on the study of mathematician at a computer company,
Overschie, Netherlands. He graduated in functions and differential equations. In she developed the Common Business-
mathematics in 1904 from the University 1947, Cartwright became the first female Oriented Language (COBOL), which
of Amsterdam, where he also taught mathematician to be elected a Fellow of became the most widely used
from 1909 to 1951. Brouwer criticized the the Royal Society in London. She had a programming language. Hopper retired
logical foundations of mathematics as long association with Girton College, from the Navy Reserve in 1966, but was
espoused by David Hilbert and Bertrand Cambridge, from 1930 to 1968, during called back on active duty the following
Russell and helped to found which time she taught, researched, year, not retiring until 1986, by which
mathematical intuitionism, based on the and served as Mistress of the college. time she held the rank of rear admiral.
view of maths governed by self-evident See also: The butterfly effect 294–99 She coined the word “bug” for a
laws. He also transformed the study of computer glitch after a moth flew into
topology by associating it with algebraic circuits on which she was working.
structures, in his fixed-point theorem. JOHN VON NEUMANN See also: The mechanical computer
See also: Topology 256–59 1903–57 222–25 ■ The Turing machine 284–89
■ 23 problems for the 20th century

266–67 ■ The logic of mathematics The son of affluent Jewish parents in

272–73 Budapest, Hungary, von Neumann was MARJORIE LEE BROWNE
a child prodigy, able to divide eight-digit 1914–79
numbers in his head at the age of six.
EUPHEMIA LOFTON HAYNES He began to publish major mathematical Only the third African-American woman
1890–1980 papers in his late teens and started to earn a PhD in maths, Browne was
teaching maths at the University of born in Tennessee at a time when it was
Born in Washington, DC, Lofton Haynes Berlin aged 24. In 1933, he moved to hard for women of colour to pursue an
was the first African-American woman the United States to take up a post academic career. With the support of her
to gain a doctorate in mathematics. at the Institute of Advanced Learning, father, a railway clerk, she graduated
After graduating from Smith College, Princeton, New Jersey, and became from Howard University, Washington
Massachusetts, with a maths degree in a US citizen in 1937. During a lifetime DC, in 1935, and, after teaching briefly
1914, she then embarked on a teaching of mathematical study, von Neumann in New Orleans, continued her studies at
career, and in 1930 established the contributed to virtually every area of the University of Michigan, gaining her
maths department at Miner Teachers the discipline. He was a pioneer of game doctorate in 1949. Two years later, she
College, which later merged with the theory, based on the “two-person zero- was appointed chair of the mathematics
University of the District of Columbia. sum game”, whereby one side wins what department at North Carolina Central
334 DIRECTORY

University. Marjorie gained a reputation American women mathematicians known as the West Area Computers,
for being an excellent teacher, and for known as the West Area Computers, comprised female African-American
her research, especially in topology. who later inspired the film Hidden mathematicians, including Katherine
See also: Topology 256–59 Figures (2016). Johnson then worked Johnson (see left). From 1958 – when
for the National Aeronautics and Space Jackson became NASA’s first female
Administration (NASA) from 1958 as black engineer – to 1963, she worked
JOAN CLARKE part of its Space Task Group. In 2015, on Project Mercury, the programme that
1917–96 President Obama awarded Johnson put the first Americans into space.
the Presidential Medal of Freedom. See also: Calculus 168–75 ■ Newton’s
London-born Clarke achieved a double See also: Calculus 168–75 ■ Newton’s laws of motion 182–83 ■ Non-Euclidean
first in maths at the University of laws of motion 182–83 ■ Non-Euclidean geometries 228–29
Cambridge on the eve of World War II geometries 228–29
but was denied a full degree because of
her gender. Her mathematical prowess ALEXANDER GROTHENDIECK
had been recognized, however, and JULIA BOWMAN ROBINSON 1928–2014
when the Bletchley Park project was 1919–85
established to decipher the German Considered by many to be the greatest
Enigma Code, Clarke was recruited. At Born Julia Bowman in St Louis, pure mathematician of the second half
Bletchley, she became one of the leading Robinson gained her mathematics of the 20th century, Grothendieck was
cryptanalysts, working closely with Alan doctorate at the University of California, unorthodox in every respect. Born in
Turing, to whom she was engaged for a Berkeley, in 1948. She developed a Germany to anarchist parents, at the
short time. Although they did the same fundamental theorem of elementary age of 10 he fled the Nazi regime as a
work as the male code-breakers, Clarke game theory (see John von Neumann, refugee to France, where he spent most
and the other Bletchley women were page 333) in 1951 but is best known for of his life. His huge output – much of it
paid less. The Bletchley Park operation her work on solving the tenth of David never published – included revolutionary
was hugely successful, cutting short the Hilbert’s list of 23 mathematical advances in algebraic geometry, the
length of the war and saving countless problems, drawn up in 1900 – whether devising of the theory of schemes, and
lives. After the war, Clarke worked at the there is an algorithm that could find a contributions to algebraic topology,
British government’s surveillance centre, solution to any Diophantine equation number theory, and category theory.
GCHQ. Because so much of Clarke’s (one that uses whole numbers and finite Grothendieck’s radical political activities
work was secret, the full extent of her unknowns). Robinson proved, along with included delivering maths lectures just
accomplishments is still unknown. other mathematicians, such as Yuri outside Hanoi while the city was being
See also: The Turing machine 284–89 Matiyasevich (see opposite), that such bombed during the Vietnam War.
■ Cryptography 314–17 an algorithm could not exist. Robinson See also: Non-Euclidean geometries
was appointed a professor at Berkeley 228–29 ■ Topology 256–59
in 1975, and in 1976 she was the first
KATHERINE JOHNSON woman to be elected to the American
1918– National Academy of Sciences. JOHN NASH
See also: Diophantine equations 1928–2015
A child maths prodigy, Katherine 80–81 ■ 23 problems for the 20th
Johnson (born Coleman) was a pioneer century 266–67 American mathematician John Nash
of computing and the American space is best known for establishing the
programme. Her calculations on flight mathematical principles of game theory
trajectories were critical in enabling MARY JACKSON (see John von Neumann, page 333).
Alan Shepard to become the first 1921–2005 After graduating from Carnegie Mellon
American in space (1961), John Glenn University in 1948 and being awarded
to be the first American to orbit Earth An aerospace engineer, Mary Jackson a doctorate from Princeton University
(1962), Apollo 11 to land on the Moon (born Winston) worked on the US space in 1950, he joined the Massachusetts
(1969), and the Space Shuttle programme programme and campaigned for better Institute of Technology (MIT), where he
to launch (1981). Johnson graduated in opportunities in engineering for women researched partial differential equations
1937 from West Virginia State College and people of colour. After graduating and began the work on game theory that
and became one of the first African- in maths and physical sciences from won him the Nobel Prize for Economics
Americans to enrol on a graduate Hampton University, Virginia, Jackson in 1994. For much of his life, Nash fought
programme at West Virignia University. taught for a while, then in 1951 started paranoid schizophrenia, as dramatized
She worked for the National Advisory work in the West Area Computing Unit in the film A Beautiful Mind (2001).
Committee for Aeronautics (NACA) of the National Advisory Committee See also: Calculus 168–75 ■ The logic
from 1953 as part of a group of African- for Aeronautics (NACA). The unit, of mathematics 272–73
 DIRECTORY 335

breakthroughs in mathematical physics, first as chair of software engineering

PAUL COHEN geometrical analysis, and topology. and later as chair of algebra and
1934–2007 A champion of gender equality in number theory.
science and mathematics, in 1990 See also: Diophantine equations
New Jersey-born Cohen was awarded she became the first woman since 80–81 ■ 23 problems for the 20th
the Fields Medal (the mathematical Emmy Noether to give a plenary century 266–67
equivalent of a Nobel Prize) in 1966 for speech at the International Congress
solving the first of David Hilbert’s list of of Mathematics. In 1994, she founded
23 unresolved mathematical problems – the Women and Mathematics Program RADIA PERLMAN
that there is no set whose number of at the Institute of Advanced Study in 1951–
elements is between that of the integers Princeton, New Jersey.
and that of the real numbers. Cohen See also: Topology 256–59 Virginia-born Perlman has been
graduated and later received his described as the “mother of the internet”.
doctorate, in 1958, from the University While a student at the Massachusetts
of Chicago before moving to the EVELYN NELSON Institute of Technology (MIT), she
Massachusetts Institute of Technology 1943–87 worked on a programme that introduced
(MIT), Princeton University, and finally children as young as three to computer
Stanford University, where he became The Krieger–Nelson Prize, awarded programming. After graduating with a
professor emeritus in 2004. by the Canadian Mathematical Society masters degree in mathematics in 1976,
See also: 23 problems for the 20th for outstanding research by a female Perlman worked for a government
century 266–67 mathematician, is named in honour of contractor that developed software.
Evelyn Nelson and fellow Canadian Then, in 1984, while working for the
Cecilia Krieger. Nelson began a career Digital Equipment Corporation (DEC),
CHRISTINE DARDEN of teaching and research at McMaster she invented the Spanning Tree Protocol
1942– University after obtaining her doctorate (STP), which ensures there is only one
there in 1970. She published more than active path between two network
Along with Katherine Johnson and Mary 40 research papers in a 20-year career devices; this would later prove crucial for
Jackson (see page 334), Darden is one of that was cut short by cancer. Her main the development of the internet. Perlman
the African-American women whose contributions were to universal algebra has taught at MIT and the universities of
work as mathematicians made key (the study of algebraic theories and their Washington and Harvard, and continues
contributions to the work of NASA’s models) and algebraic logic, applying to work on computer network and
space programmes. After graduating these to the field of computer science. security protocols.
from Hampton University, Darden taught See also: The fundamental theorem See also: The mechanical computer
at Virginia State University before of algebra 204–09 ■ The logic of 222–25 ■ The Turing machine 284–89
moving in 1967 to NASA’s Langley mathematics 272–73
Research Center. There, she built her
reputation as an aeronautical engineer, MARYAM MIRZAKANI
specializing in supersonic flight. In 1989, YURI MATIYASEVICH 1977–2017
she was appointed leader of the Sonic 1947–
Boom Team, working on designs to At the age of 17, Mirzakani became the
reduce noise pollution and other While studying for his doctorate at the first Iranian woman to win a gold medal
negative effects of supersonic flight. Steklov Institute of Mathematics in in the International Mathematical
See also: Calculus 168–75 ■ Newton’s Leningrad (now St Petersburg), Olympiad. She graduated from Tehran’s
laws of motion 182–83 ■ Non-Euclidean Matiyasevich became fascinated by Sharif University of Technology, before
geometries 228–29 the challenge of solving David Hilbert’s earning a doctorate from Harvard
tenth problem. Just as he was about to University in 2004 and taking up a
give up, he read the paper “Unsolvable professorship at Princeton University.
KAREN KESKULLA UHLENBECK Diophantine problems” (1969) by Ten years later, Mirzakani was both the
1942– American mathematician Julia first woman and the first Iranian to
Robinson (see page 334), and a solution receive the Fields Medal – for her
In 2019, Uhlenbeck became the first fell into place. In 1970, Matiyasevich contribution to the study of Riemann
woman to be awarded the Abel Prize provided the final proof that the tenth surfaces. She was working at Stanford
for Mathematics. Born in Cleveland, problem is unsolvable because there University when she died of breast
Ohio, in 1942, she gained a PhD in is no general method of determining cancer, aged 40.
mathematics from Brandeis University, whether Diophantine equations have See also: Non-Euclidean geometries
Waltham, Massachusetts in 1968, a solution. In 1995, he was appointed 228–29 ■ The Riemann hypothesis
and went on to achieve notable professor at St Petersburg University, 250–51 ■ Topology 256–59
336

GLOSSARY
In this glossary, terms defined Analysis The branch of Base (1) In a number system,
within another entry are identified mathematics that studies limits the base is the number around
with italic type. and handles infinitely large and which the system is organized.
small quantities, especially to The main number system we use
Abstract algebra The branch of solve problems in calculus. today is the base-10 or decimal
algebra, developed mainly in the system, where the numerals 0 to 9
20th century, that investigates Analytic geometry See are used and the next number is
abstract mathematical structures algebraic geometry. written 10, indicating one ten
such as groups and rings. and no units. See also place
Apex The vertex that is furthest value system. (2) In logarithms, a
Acute angle An angle that is less from the base in a 3-D shape. fixed base (usually 10 or Euler’s
than 90 degrees. number e) is used; the logarithm
Applied mathematics The use of of any given number x is the power
Algebra A branch of mathematics mathematics to solve problems in to which that base must be raised
that involves the use of letters to science and technology. It includes to produce x.
stand for unknown or variable techniques for solving particular
numbers in calculations. kinds of equations. Binary notation Writing numbers
using the binary system, in which
Algebraic geometry The use Arc A curved line that forms part the only numerals used are 0 and 1.
of graphs to plot lines and curves of the circumference of a circle. For example, the number 6 is
that represent algebraic functions, written as 110 in the binary system.
such as y = x2. Area The amount of space inside Here, the leftmost 1 has the value of
any 2-D shape. Area is measured 4 (2  2), the middle 1 means one 2,
Algebraic numbers All the in square units, such as square and the zero means no single units:
rational numbers and those centimetres (cm2). 4 + 2 + 0 makes 6.
irrational numbers that can
be obtained by calculating the Associative law This states that Binomial An expression
roots of a rational number. An if you add, for example, 1 + 2 + 3, consisting of two terms added
irrational number that is not the numbers can be added in any together, e.g. x + y. When a
algebraic (such as pi or e) is order. The law works for ordinary binomial expression is raised to
called a transcendental number. addition and multiplication, but a power, for example ( x + y)3, the
not for subtraction or division. result when multiplied out gives
Algorithm A defined sequence of (in this case) x3 + 3x2y + 3xy2 + y3.
mathematical or logical instructions, Average The typical or middle This process is called binomial
or rules, devised to solve a class of value of a set of data. For the expansion, and the numbers
problems. Algorithms are widely different kinds of averages, see multiplying the terms (3s in
used in mathematics and computer mean, median, and mode. this case) are called binomial
science for calculation, organizing coefficients. The binomial theorem
data, and a multitude of other tasks. Axiom A rule, especially one is a rule for working out binomial
that is fundamental to an area coefficients in complex cases.
Amicable numbers Any pair of of mathematics. See also polynomial.
whole numbers, where the factors
of each one add up to form the Axis (plural axes) A fixed reference Calculus A branch of mathematics
other. The smallest pair are 220 line, such as the vertical y-axis and that deals with continuously
and 284. horizontal x-axis on a graph. changing quantities. It includes
 GLOSSARY 337

differential calculus, which is Complex number A number that Cosine (abbreviation cos) A
concerned with rates of change, is a combination of a real number function in trigonometry similar to
and integral calculus, which and an imaginary number. a sine, except that it is defined as
calculates areas and volumes the ratio of the length of the side
under curves or curved surfaces. Complex plane The infinite 2-D of a right-angled triangle adjacent
plane on which complex numbers to a given angle to the length of the
Cardinal numbers Numbers that can be plotted. triangle’s hypotenuse.
denote a quantity, such as 1, 2, 3
(in contrast to ordinal numbers). Composite number A whole Cube A 3-D geometrical figure
number that is not prime but can whose faces are six identical
Chord A straight line that cuts be created by multiplying together squares. A cube number is one
across a circle but does not go smaller numbers. that is obtainable by multiplying a
through its centre. smaller number together twice –
Cone A 3-D shape with a circular for example 8, which is 2  2  2
Cipher Any systematic method base and a side that narrows (23). This multiplication resembles
of coding messages so that they upwards towards a point (apex). the way the volume of a cube is
cannot be understood without calculated, by multiplying its
being deciphered first. Congruent Having the same size length  height  depth.
and shape. (Used when comparing
Circumference The distance geometrical shapes.) Cubic equation An equation
all the way round the outside edge containing at least one variable
of a circle. Conjecture A mathematical multiplied by itself twice (for
statement or claim that has not yet example, y  y  y, also written as
Coefficient A number or been proved or disproved. A pair of y3), but no variable multiplied more
expression, usually a constant, related conjectures can be strong times than this.
that is placed before another or weak: if the strong conjecture is
number (especially a variable) proved then the weak conjecture Cubit A measure of length used
and multiplies it. For example, is also proved, but not vice versa. in the ancient world, based on
in the expressions ax2 and 3x, the length of the human forearm.
a and 3 are coefficients. Constant A quantity in a
mathematical expression that does Cylinder A 3-D shape, such as a
Coincident In geometry, two or not vary – often symbolized by a tin can, with two identical circular
more lines or figures that, when letter such as a, b, or c. ends joined by one curved surface.
superimposed on each other, share
all points and occupy exactly the Convergence A property of some Deduction A process by which a
same space. infinite mathematical series where problem is solved by drawing on
not only is each term smaller than known or assumed mathematical
Combinatorics A branch of the last but the terms, when added principles. See also induction.
mathematics that studies the ways up, approach a finite answer. The
in which sets of numbers, shapes, value of numbers such as pi can be Degree (1) A measure of angle
or other mathematical objects can estimated using convergent series. in geometry: rotating a full circle
be combined. involves turning 360 degrees.
Coordinates Combinations of (2) The degree or order of a
Commutative law The law that numbers that describe the position polynomial is the highest-power
states that 1 + 2 = 2 + 1, for of a point, line, or shape on a graph term within it: for example, a
example, and that the order in or a geographical position on a map. polynomial is “of degree 3” or
which the numbers are set down In mathematical contexts, they are “of order 3” if it contains a cubed
doesn’t matter. It works for ordinary written (for a 2-D case) in the form term, such as x3, as its highest
addition and multiplication, but not (x, y), where x is the horizontal power. Similarly, in differential
for subtraction and division. position and y the vertical position. equations, the term that has
338 GLOSSARY

been differentiated most times quantities are equal to each other. Factorial The product of any
in a given equation determines An equation is the usual way positive integer and all the positive
its degree or order. of expressing a mathematical integers that are smaller than it. For
function. When an equation is example, factorial 5, also written 5!
Denominator The lower number true of all the values of a variable (with an exclamation mark) is 5  4
in a fraction, such as the 4 in ¾. (for example, the equation y  y   3  2  1 = 120.
y = y3), it is called an identity.
Derivative See differentiation. Factorization Expressing a
Equilateral triangle A triangle number or mathematical expression
Diameter A straight line touching in which all three sides are the in terms of factors that when
two points on the edge of a circle same length and all three angles multiplied together give the
and passing through the centre. the same size. original number or expression.

Differential equation An Existence proof A mathematical Formula A mathematical rule

equation that represents a function proof that something exists, that describes a relationship
including the derivative(s) of a obtained either by constructing an between quantities.
given variable. example or by general deduction.
Fractals Self-similar curves or
Differentiation In calculus, the Expansion In algebra, the shapes of different sizes that form
process of working out the rate of expansion of an expression is complex patterns that have the
change of a given mathematical the opposite of factorization. same general appearance at any
function. The result of the For example, (x + 2)(x + 3) can magnification. Many natural
calculation is another function be expanded to x2 + 5x + 6, by phenomena, such as clouds and rock
called the differential or derivative multiplying each term in the formations, approximate to fractals.
of the first function. first pair of brackets by each term
in the second pair of brackets. Function A mathematical
Divergence A term applied relationship where the value of one
mainly to infinite series that do not Exponent The superscript number variable is worked out uniquely
approach closer and closer to an that indicates the power to which a from the value of other numbers,
end-number. See also convergence. number or quantity has been using a particular rule. For
raised, such as the 2 in x2 ( x  x). example, in the function y = x2 + 3,
Divisor The number or quantity by Also called an index. the value of y is calculated by
which another number or quantity squaring x and then adding 3. The
is being divided. Exponential function A same function can also be written
mathematical function where, as f (x) = x2 + 3, where f (x) stands for
Dodecahedron A 3-D polyhedron a quantity gets larger, its rate of “function of x”.
made up of 12 pentagonal (5-sided) increase also gets faster. The result
faces. A regular dodecahedron is is often called exponential growth. Geometry The branch of
one of the five Platonic solids. mathematics that studies
Expression Any meaningful shapes, lines, points, and
Ellipse A shape like a circle but combination of mathematical their relationships. See also
stretched out symmetrically in symbols, such as 2x + 5. non-Euclidean geometries.
one direction.
Face A flat surface of any 3-D shape. Gradient The slope of a line.
Encryption The process of
converting data or a message Factor A number or expression Graph (1) A chart on which data is
to a secure, coded form. that divides exactly into another plotted using, for instance, lines,
number or expression. For example, points, curves, or bars. (2) In graph
Equation A statement that two 1, 2, 3, 4, 6, and 12 are all factors theory, a graph is a collection of
mathematical expressions or of 12. points, called vertices, and lines,
 GLOSSARY 339

called edges, that can be used to Identity element In a set of as involving the adding up of
model theoretical and real networks, numbers or other mathematical infinitesimals (infinitely small but
relations, and processes in a range of objects, an operation carried out non-zero quantities).
scientific and social fields. on the set, such as multiplication
or addition, always has an identity Input Any variable, which when
Graph theory A branch of element – a number or expression combined with a function,
mathematics that studies how that leaves other terms unchanged produces an output.
graphs made up of points and after the operation has been
lines are connected. carried out. The identity element Integer Any of the negative or
in ordinary multiplication, for positive whole numbers. (Fractions
Group A mathematical set, example, is 1, as 1  x = x, and in are not integers.)
together with an operation which, the addition of real numbers, it is
when performed on members of the 0, as 0 + x = x. Integral (1) Relating to integers.
set, yields an answer that is still a (2) A mathematical expression used
member of the set. For example, the Imaginary number Any number in integral calculus, or the result of
set of integers forms a group when that is a multiple of -1, which performing an integration.
addition is the operation. Groups does not exist as a real number. It
can be finite or infinite, and their is expressed as the symbol i. Integration The process of
study is called group theory. performing a calculation in
Incommensurable Something integral calculus.
Harmonic series The that cannot be measured exactly
mathematical series 1 + 1 ⁄2 + 1 ⁄3 + in terms of something else. Inverse A mathematical expression
1 ⁄4 + 1 ⁄ 5 +… . The individual terms or operation that is the opposite of
in the series define the different Index (plural indices) Another another one and undoes it. For
ways that a stretched string, for word for an exponent. example, division is the inverse
example, or air in a tube, can vibrate of multiplication.
to produce sound. The resulting Induction A way of obtaining a
series of musical pitches forms the general conclusion in mathematics Irrational number Any number
basis of the musical scale. by establishing that if a statement that cannot be expressed as one
is true for one step in a process it whole number divided by another
Hyperbola A mathematical curve is also true for the next step in a and is not an imaginary number.
that looks something like a process and all those that follow.
parabola, but in which the two See also deduction. Isosceles triangle A triangle with
extensions of the curve approach two sides the same length and two
two imaginary straight lines at Infinite Indefinitely large and angles the same size.
angles to each other without ever without limit. In mathematics, there
touching or crossing the lines. are different kinds of infinity: the set Iteration Performing the same
of natural numbers, for example, operation again and again to
Hypotenuse The longest side of a is countably infinite (countable one achieve a desired result.
right-angled triangle, located on the by one, even though the end is never
opposite side from the right angle. reached), while the real numbers are Limit The end number that is
uncountably infinite. approached as certain calculations
Icosahedron A 3-D polyhedron are iterated to infinity.
made up of 20 triangular faces. Infinite series A mathematical
A regular icosahedron is one of series with an infinite number of Linear equation An equation
the five Platonic solids. terms: see series. that contains no variable
multiplied by itself (for example,
Ideal In abstract algebra, a Infinitesimal calculus Another no x2 or x3). Linear equations
mathematical ring that is term for calculus, generally used in result in straight lines when
a component of a larger ring. the past when calculus was viewed they are plotted as graphs.
340 GLOSSARY

Linear transformation Also describing vectors, calculating left, the highest on the right.
called linear mapping, a mapping transformations in the shape and All real numbers can be placed
between vector spaces. position of geometrical figures, and on a number line.
representing real-world data.
Logarithm The logarithm of a Number system Any system
number is the power to which Mean An average found by adding of writing down and expressing
another number (called the up the values of a set of data and numbers. The Hindu–Arabic system
base – usually either 10 or Euler’s dividing by the number of values. used today is based on the ten
number e) – must be raised to give For example, the mean of the four numerals 0 to 9: when 10 is reached,
the original number. For example, numbers 1, 4, 6, and 13 is 1 + 4 + 6 1 is written again, but with a 0
100.301 = 2, and so 0.301 is the + 13 = 24 divided by 4 = 6. after it. This system is both a
logarithm (to the base 10) of 2. place value system and a base-10
A logarithm to the base e Median The middle value of a set or decimal system.
(2.71828… ) is called a natural of data, when the values are put in
logarithm and is indicated by the order from lowest to highest. Number theory A branch of
prefix ln or loge. The advantage of mathematics that studies the
logarithms is that when numbers Meridian An imaginary line on properties of numbers (especially
need to be multiplied, the Earth’s surface joining the North whole numbers), their patterns,
calculation can be simplified by Pole and South Pole through any and their relationships. It includes
adding their logarithms instead. given locality. Lines of longitude the study of prime numbers.
are meridians.
Logic The study of reasoning, Numerator The upper number
that is, how conclusions can be Mode The value that occurs most in a fraction, such as the 3 in ¾.
deduced correctly from given often in a set of data.
starting information (premises) Obtuse angle An angle between
by following valid rules. Modular arithmetic Also 90 and 180 degrees.
called clock arithmetic, a form of
Manifold A kind of abstract arithmetic where, after counting Octahedron A 3-D polyhedron
mathematical space that in any up to a certain point, 0 is reached, made up of eight triangular faces.
particular small region resembles and the process is repeated. A regular octahedron is one of the
ordinary 3-D space. It is a concept five Platonic solids.
within topology. Natural logarithm See logarithm.
Operation Any standard
Mapping Establishing a Natural number Any of the mathematical procedure such as
relationship between members of positive whole numbers. See addition or multiplication. The
one mathematical set and another. also integer. symbols used for such operations
It is often but not always used to are called operators.
mean a one-to-one mapping, Non-Euclidean geometries A key
where each member of one set is postulate of traditional geometry, as Order See degree.
associated with one member of described by Euclid in ancient
the other set, and vice versa. times, is that parallel lines never Ordinal numbers Numbers that
meet (often expressed as meeting at denote a position, such as 1st, 2nd,
Matrix (plural matrices) A square infinity). Geometries in which this or 3rd. See also cardinal numbers.
or rectangular array of numbers or and other Euclidean postulates are
other mathematical quantities that not valid are called non-Euclidean. Origin The point at which the x
can be treated as a single object in and y axes of a graph intersect.
calculations. Matrices have special Number line A horizontal line
rules for addition and multiplication. with numbers written on it that is Oscillation A regular to-and-fro
Their many uses include solving used for counting and calculating. movement between one position or
several equations simultaneously, The lowest numbers are on the value to another and back again.
 GLOSSARY 341

Output The result when an input Placeholder A numeral, usually four ys multiplied together (y  y 
is combined with a function. zero, used in a place value system to y  y) is called “y raised to the
differentiate 1 from 100, for example, power of 4” and written y4.
Parabola A curve that is similar but that does not necessarily imply
to one end of an ellipse, except that an exact measurement as in phrases Power series A mathematical
the arms of a parabola diverge. such as “about 100 km away”. series where each term has a
greater power than the previous
Parabolic Relating to a parabola, Plane A flat surface. one, such as x + x2 + x3 + x4 +… .
or to a function based on it, such
as a quadratic function, which Plane geometry The geometry Prime number A natural number
produces a parabola-shaped graph. of 2-D figures on a flat surface. that can be divided exactly only by
itself and 1.
Parallel Of a line, going in exactly Platonic solid One of the five
the same direction as another line. polyhedra that form completely Probability The branch of
regular and symmetrical shapes: mathematics that studies the
Parallelogram A quadrilateral each face is an identical polygon likelihood of different outcomes
where each side has the same and all the angles between the occurring in the future.
length as the side opposite to it and faces are the same. The five
the two sides are also parallel. A Platonic solids are the tetrahedron, Product The result of one
square, rectangle, and rhombus cube, octahedron, dodecahedron, number or quantity being
are types of parallelogram. and icosahedron. multiplied by another.

Partial differential equation Polygon Any flat shape with three Proof Any method of showing
A differential equation containing or more straight sides, such as a beyond doubt that a mathematical
several variables, in which the triangle or pentagon. statement or result is true. There
differentiation is applied to only are different kinds, including proof
one of the variables at a time. Polyhedron Any 3-D shape whose by induction and existence proofs.
faces are polygons.
Periodic function A function Proportion The relative size
whose value repeats periodically, Polynomial A mathematical of something compared with
as seen, for example, in the graph expression made up of two or something else. For example,
of a sine function, which is in the more terms added together. A if two quantities are in inverse
form of a repeating series of waves. polynomial expression usually proportion, the larger one of them
includes different powers of a becomes, the smaller the other one
Perpendicular At right angles to variable, together with constants, will become; for example, if one
something else. for example, x3 + 2x + 4. quantity is multiplied by 3, the
other is divided by 3.
Pi () The ratio of a circle’s Positional number An individual
circumference to its diameter, numeral whose value depends on Pure mathematics Topics in
approximately 22 ⁄ 7, or 3.14159. It is its position within a larger number. mathematics that are studied
a fundamental transcendental See place value system. for their own sake rather than for
number that appears in many any practical application. See also
branches of mathematics. Postulate In mathematics, a applied mathematics.
statement whose truth is taken for
Place value system The standard granted or thought to be obvious, Quadratic equation An equation
system for writing numbers, where but is not supported by a proof. containing at least one variable
the value of a digit depends on its multiplied by itself once (for
place in a larger number. The 2 in Power The number of times a example y  y, also written y2),
120, for example, has a place value quantity or number has been but containing no variables raised
of 20, but in 210 it stands for 200. multiplied by itself. For example, to higher powers.
342 GLOSSARY

Quadrilateral Any flat 2-D shape result of multiplying them together Segment (1) Part of a line, with
with four straight sides. is 1. For example, the reciprocal definite end points. (2) In a circle,
of 3 is 1 ⁄3. the area between a chord and the
Quartic Referring to equations outside edge (circumference).
or expressions of the fourth degree, Recurring Any number that
where the highest power contained is repeated without limit. For Sequence An arrangement of
in them is 4 – for example, x4. example, 1 ⁄3 expressed in decimals numbers or mathematical terms
is 0.333333… , which can also be placed one after the other and
Quaternion A mathematical described in words as “nought usually following a set pattern.
object that is a development of the point three recurring”.
idea of a complex number but uses Series A list of mathematical
four components added together, Rhombus A quadrilateral with terms added together. Series
rather than just two. all four sides the same length; usually follow a mathematical rule,
informally, a diamond shape. A and even if the series is infinite,
Quintic Referring to equations or square is a special kind of rhombus, it may add up to a finite number.
expressions of the fifth degree, with all angles 90 degrees. See also sequence.
where the highest power contained
in them is 5, for example, x5. Right angle An angle that is Set Any collection of numbers,
90 degrees (a quarter turn), such or mathematical structures based
Quotient The result that is as the angle between vertical on numbers. Sets can be finite
obtained when one number is and horizontal lines. or infinite (for example, the set of
divided by another. all integers).
Ring A mathematical structure
Radian A measure of angles that is that is like a group except that it Set theory The theory of sets and a
an alternative to degrees and is includes two operations rather branch of mathematics which now
based on the length of the radius and than one. For example, the set of forms the underlying basis of many
circumference of a circle. Turning all integers forms a ring when other branches of mathematics.
around by 2  pi (2) radians is the taken together with the operations
same as turning 360 degrees (that addition and multiplication, Sexagesimal A number system
is, in a complete circle). because performing these used by the ancient Babylonians
operations on members of the based on the number 60, and still
Radius Any straight line from set produces an answer that is used in a modified form for time,
the centre of a circle or sphere still a member of the set. angles, and geographic coordinates.
to its circumference.
Root (1) The root of a number, Simultaneous equations A set
Rational number A number that which is another number that of several equations that include
can be expressed as a fraction of when multiplied gives the original the same unknown quantities, such
one whole number over another. number. For example, 4 and 8 are as x, y, and z. Usually, the equations
See also irrational number. roots of 64, with 8 being the square must be calculated together to
root (8  8 = 64) and 4 the cube solve the value of the unknowns.
Real number Any number that root (4  4  4 = 64). (2) The root
is either a rational number or an of an equation is its solution. Sine (abbreviation sin) An
irrational number. Real numbers important function in trigonometry,
include fractions and negative Scalar A quantity that has and defined as the ratio of the
numbers, but not imaginary or magnitude (size), but not direction, length of the side opposite a given
complex numbers. in contrast to a vector. angle in a right-angled triangle
to the length of the triangle’s
Reciprocal A number or Scalene triangle A triangle where hypotenuse. This ratio starts at
expression that is the inverse of none of the sides and none of the 0 and varies with the size of the
another one, meaning that the angles are the same size. angle, repeating its pattern after
 GLOSSARY 343

360 degrees. The graph of a sine regular geometrical shapes that Translation A function that
function is also the shape of many cover the surface without any moves an object a certain distance
waves, including light waves. gaps in between. This is also in a direction without affecting
called a tiling. its shape, size, or orientation.
Slope The angle of a line to the
horizontal, or an angle of a tangent Tesseract A 4-D shape with four Trigonometry Originally, the
to a curve to the horizontal. edges at every vertex, whereas a study of the way the ratios between
cube has three edges at every different sides of a right-angled
Square number A whole number vertex, and a square has two. triangle change when other angles
that can be formed by multiplying a in the triangle change, and later
smaller whole number by itself Tetrahedron A 3-D polyhedron extended to all triangles. The way
once. For example, 25 is a square that is made up of four triangular the ratios change is described by
number as it is 5  5 (52). faces. A regular tetrahedron is trigonometric functions, which are
one of the five Platonic solids. now fundamental to many branches
Statistics (1) Measurable data of mathematics.
collected in an orderly way for Theorem A significant proven
any purpose. (2) The branch of result on a mathematical topic, Variable A mathematical quantity
mathematics that develops and especially one that is not self- that can take on different values,
applies methods for analysing evident. An unproved statement often symbolized by a letter such
and studying such data. is called a conjecture. as x or y.

Surd An expression that includes Topology The branch of Vector A mathematical or

a root that is an irrational number mathematics that studies surfaces physical quantity that has both
such as 2. It is left in root form as and objects by examining how magnitude and direction. In
it cannot be simplified or written their parts are connected rather diagrams, vectors are often
exactly as a decimal. than according to their exact represented by bold arrows.
geometrical shapes. For example,
Surface area The area of a flat a doughnut and a teacup are Vector space A complex
or curved surface, or of the outside topologically similar because abstract mathematical structure
of a 3-D object. they are both shapes that have that involves the multiplication
one hole going through them of vectors by each other and
Tangent (1) A line which grazes (going through the handle, in by scalars.
the outside of a curve, just touching the case of the teacup).
it at one point. (2) In trigonometry, Venn diagram A diagram that
the tangent function, abbreviated Transcendental number Any shows sets of data as overlapping
as tan, is defined as the ratio of the irrational number that is not an circles. The overlaps show what
side length opposite a given angle algebraic number. The number pi the sets have in common.
to the side length adjacent to that () and Euler’s number e are both
angle, in a right-angled triangle. transcendental numbers. Vertex (plural vertices) A corner
or angle, where two or more lines,
Term In an algebraic expression, Transfinite number Another curves, or edges meet.
one or more numbers or variables, term for an infinite number. It is
usually separated by a plus () or used particularly when infinities Volume The amount of space
minus (-) sign, or in a sequence, of different sizes or infinite inside a 3-D object.
by a comma. In x  4y - 2, for collections of objects are compared.
example, x, 4 y, and 2 are all terms. Whole number Any of the
Transformation The conversion negative and positive counting
Tessellation A pattern that is of a given shape or mathematical numbers. For example, –1, 0,
formed on a flat 2-D surface by expression into another related one, 19, 55, and so on. It is another
repeated copies of one or more using a particular rule. term for integer.
344

INDEX
Page numbers in bold refer to main entries; Möbius strip 212, 248–49, 258 Bell, Eric Temple 233
those in italics refer to illustrations and captions. Penrose tiling 305 Beltrami, Eugenio 229
projective geometry 116, 154–55 Benford, Frank 290
applied mathematics 264, 280 Benford’s law 290

A Fourier analysis 216–17

laws of motion 182–83, 218
Arbuthnot, John 192
Berlin papyrus 28, 29, 29, 32
Bernoulli, Daniel 216, 219, 221
Bernoulli, Jacob 122, 158, 165, 167, 174, 180,
Archimedes 62–63, 63, 69, 83, 99, 102, 104, 112, 185, 185, 188, 189–90, 192, 198
abacus 24, 26–27, 58–59, 59, 87 142, 152–53, 170, 171 Bernoulli, Johann 174, 180, 185, 188, 190
abax counting board 89 arctangent series 62, 65
Abel, Niels Henrik 102, 200, 226, 227 Bernoulli trial 192
Argand, Jean-Robert 128, 146, 206, 209, 214,
Abel Prize 323 Bessel, Friedrich Wilhelm 221
215, 234
Abelian groups 233 Argand diagrams 128, 215, 215 Bessel functions 221
abstract algebra 16, 98, 213, 264, 280–81 Aristotle 21, 46, 47, 50–51, 51, 172, 182, 216, Bhaskara II 74
Abu Kamil Shuja ibn Aslam 28, 30, 96–97 244, 278, 300 binary system 27, 176–77, 265, 289, 291, 300
Achilles and the tortoise paradox 46–47, 47 arithmetic 15, 20 Binet, Jacques 241
addition 14 see also addition; division; multiplication; binomial coefficients 98, 101, 101, 159–60
symbol 79, 126, 127 number systems; number theory; subtraction binomial distribution 192–93
Adelard of Bath 64 arrow paradox 46 binomial theorem 87, 100–01, 203
Adleman, Leonard 314 art Bitcoin 314
Age of Enlightenment 180–81 fractal art 311, 311 black holes 274
Aglaonike of Thessaly 82 golden ratio in 120, 122, 122, 123 Black Sabbath 293
Agnesi, Maria 82 mathematician–artists 229, 249 Blake, William 311
Alberti, Leon Battista 121, 154, 155 artificial intelligence (AI) 113, 297, 301 Boëthius 59
Aleksandrov, Pavel 256 machine learning 301 Bollobás, Béla 304
Alembert, Jean d’ 191, 206, 208, 208, 279 Turing test 265, 289 Boltzmann, Ludwig 218, 219, 291
algebra 15–16 Artin, Emil 302 Bolyai, János 54, 212, 229
abstract algebra 16, 98, 213, 264, 280–81 Aryabhata 64, 73–74 Bolyai, Wolfgang 229
binomial theorem 87, 100–01, 203 Aschbacher, Michael 318 Bombe 287, 289, 316
Boolean algebra 176, 213, 242–47, 286, 291, astrolabes 73
300, 301 Bombelli, Rafael 75, 105, 117, 128–29, 130–31,
astronomy 73, 74–75, 87, 89, 90, 143, 219, 221, 226
Bourbaki group 264, 282–83 207, 234
see also planets
equations see equations Bombieri, Enrico 251
asymmetric encryption 317
fundamental theorem of algebra (FTA) 200, Atiyah, Michael 251 Boole, George 50, 244–47, 245, 265
201, 204–09, 215 Automatic Computing Engine (ACE) 287 Boole, Mary Everest 244
group theory 200, 201, 213, 230–33, 240 automorphic forms 302, 303 Boolean algebra 176, 213, 242–47, 286, 291,
matrices 238–41 axioms 51, 55, 57, 68, 231, 232–33, 286 300, 301
notation 126–27, 146 axis of perspectivity 154 Boolean commands 247
origins of 92–99 Aztecs 59 Borel, Émile 278, 279, 279
symbolic algebra 21, 94, 99 Bortkiewicz, Ladislaus 220
algebraic geometry 283, 323 Bourbaki, Nicolas see Bourbaki group

B
algebraic topology 259, 282 Bourbaki group 264, 282–83
algorithms 62–63, 77, 89, 225, 286, 288, 289, 315 Bouyer, Martin 65
Eratosthenes’ sieve 66–67, 286–87 Bovelles, Charles de 152, 153, 167
Fast Fourier Transform (FFT) 216 Boyer, Carl Benjamin 74
halting problem 288 Babbage, Charles 213, 222, 223–24, 223, 225, 286 brachistochrone problem 167
iterative algorithms 83 Babylonian mathematics 20, 24, 25–26, 29–30, Brahmagupta 24, 28, 30, 64, 74, 76, 78, 86, 88,
RSA algorithm 314–15, 317 39, 44, 58, 59, 62, 66, 72, 73, 88, 89, 94, 102,
89, 90–91, 95, 99, 200
allometric scaling 308 134, 170, 200
Brahmagupta’s formula 74
analytic geometry 69, 117, 148, 149 Bacon, Francis 177
Bravais, Auguste 269
analytical calculus 151 Bacon, Kevin 293, 304
Analytical Engine 222–23, 224–25, 286 barber paradox 252, 272, 273 Breakthrough Prize 196
Anaximander 39 Barr, Mark 120 Briggs, Henry 140–41
Anderson, Jesse 278 Barrow, Isaac 171, 173 Browne, Sir Thomas 41
Apollonius of Perga 68–69, 82, 95, 146, 150, 154 al-Battani (Albatenius) 74, 172 Brunelleschi, Fillipo 155
Appel, Kenneth 313 Baudhayana 80 Buffon, Georges-Louis Leclerc, Comte de 181,
applied geometry 36–43 Bayes, Thomas 181, 184, 199 202–03, 203
Bessel functions 221 Bayes’ theorem 198–99 Buffon’s needle experiment 202–03
cycloids 152–53, 167 beauty, ratio of 123, 123 builder’s square 14–15
golden ratio 110, 111, 116, 118–23 bell curve 184, 192–93, 192, 198–99, 268–69 butterfly effect 296–97, 310
 INDEX 345

Clarke, R. D. 220

C classical mechanics 218, 219

classification circles 254
classification theorems 55
D
Clay Millennium Prize 265, 266, 267,
cabtaxi numbers 277 324, 325 Dalí, Salvador 123
Caesar cipher 315 climate modelling 296–98, 310 De la Vallée Poussin, Charles-Jean 260
calculators, mechanical 58, 177 Cohen, Paul 267 De Morgan, Augustus 78, 244, 313
calculus 16, 47, 63, 100, 117, 143, 168–75, 180, coin toss 185, 185, 192, 198, 199, 279 De Sola Pool, Ithiel 292
181, 185, 218, 272, 298 Colburn, Zerah 235 Debussy, Claude 111
analytical calculus 151 collinearity 155 decimal currency 137
differential calculus 16, 82, 173, 174 Colossus computer 222 decimal point 24, 27, 116, 136
fundamental theorem of calculus 170, combinatorics 158, 237, 304 decimal system 24, 26, 27, 32, 89, 96, 132–37
173, 175 commutative rings 281 notation 135–37, 135
infinitesimal calculus 47, 126 completing the square 30, 97, 97 see also Hindu–Arabic numeration
integral calculus 16, 82, 170, 173 complex analysis 131, 209, 261 decimal time system 137
Leibnizian 117, 142, 143, 174, 185 complex functions 214, 251, 279 Dedekind, Richard 280
Newtonian 100, 117, 126, 143, 171, 173–74, complex logarithms 197 deductive reasoning 21, 51
185, 199, 218, 298 complex numbers 75, 105, 128–31, 150, 193, Dehn, Max 267
notation 175 197, 200, 207, 209, 233, 234 Deligne, Pierre 276
calendar Argand diagrams 128, 215, 215 delimiters 136–37
Babylonian 25 complex plane 209, 214–15, 234, Delsarte, Jean 283
Mayan 27 235, 310 derangements 191
Persian/Jalali 103, 105 quaternions 146, 151, 212, 214, Desargues, Girard 116, 154–55
Cantor, George 16, 213, 252–53, 253, 266, 267, 234–35 Descartes, René 69, 78, 99, 105, 117, 122, 127,
282, 308–09 see also imaginary numbers 146, 153, 165, 173, 214, 282, 283
Cantor set 309, 310 composite numbers 66, 67 see also Cartesian coordinates
carbon-dating 191 compound interest 185, 188, 189–90 determinants 240–41
Cardano, Gerolamo 8, 28, 30, 91, 105, 117, 128, computer aided design (CAD) 155, 241 determinism 212, 218, 219, 296, 298, 299
130, 158, 162, 165, 184, 200, 202, 206, 207, computer circuitry 247, 286 Devlin, Keith 123, 215
214, 215, 238 computer science 17, 247, 265 dialectical method 47, 50
Cartan, Henri 283, 283 binary system 176–77 dichotomy paradox 46
Cartesian coordinates 117, 144–51, 195, cryptography 241, 265, 314–17 Dieudonné, Jean 283
258, 282 first programmable computer 225, 289 Difference Engine 213, 222, 223, 224, 225
cartography 67, 75, 155 information theory 289, 291 differentiation 170, 173, 175
four-colour theorem 312–13 mechanical computer 222–25 digital file compression 217
Cassels, J. W. S. 237 proof by computers 313 Diophantine equations 80–81
Catalan, Eugène 213, 236–37, 237 quantum computing 289 Diophantus 21, 80–81, 82, 94, 95, 100, 126, 130,
Catalan’s conjecture 236–37 search engines 241, 247 196, 206, 320
catenaries 190 Turing machine 284–89 divination 34, 35, 177
catenoids 190 von Neumann architecture 289 Divine Comedy (Dante) 43
Cauchy, Augustin-Louis 40, 170, 173, 174, 175, conic sections 55, 68–69, 103, 104, 154, Divine Proportion see golden ratio
230, 241, 272 155, 226 division 14, 137
Cavalieri, Bonaventura 152, 172–73 conservation laws 233, 280 symbol 126, 127
Cayley, Arthur 230, 238, 239, 241, 280 continuous function 308 zero and 91
Cayley graph 319, 319 continuum hypothesis 266, 267 DNA 259
celestial coordinates 151 control theory 241 dodecahedra 41, 48, 49, 49, 235
central limit theorem 184 Cook, Stephen 286 double false position 77
CERN accelerator 233 Cooley, James 216 Dresden Codex 27
Ceulen, Ludolph van 65 correlation 270, 271 Dreyfus, Alfred 261
chaos theory 16, 257, 265, 283, 296–99, 310 Cotes, Roger 197 Drinfel’d, Vladimir 303
Chebyshev, Pafnuty 250 Cournot, Antoine Augustin 278, 279 du Sautoy, Marcus 27
Chevalley, Claude 283 Cournot’s principle 279 Durand–Kerner method 209
chi-squared test 268, 271 Cramer, Gabriel 241 Dürer, Albrecht 35
Chinese mathematics 21, 24, 26, 34, 39, 76–77, Cramer’s rule 241 dynamical systems 298, 309–10
83, 239 cryptocurrency 314
see also specific mathematicians cryptography 177, 226, 241, 265, 314–17

E
Chudnovsky, David and Gregory 65 public key encryption 67, 227, 233,
Chuquet, Nicholas 127, 138–39 317, 317
Cicero 278 cubes 48, 49, 49, 234
cipher wheels 315 cubic equations 78, 87, 91, 98, 99, 102–05, 117,
ciphers see cryptography 128, 129–30, 200, 207, 215 e see Euler’s number
ciphertext 315, 316, 317 Cuming, Alexander 192 Eddington, Arthur 233, 278
circles 33, 68, 69, 69, 149, 170, 171, 226–27, cuneiform characters 20, 25, 38 Egyptians, ancient 20, 24, 29, 32–33, 39, 73, 94,
254 cyclic quadrilaterals 74 134, 170, 176
nine-point circle 166 cycloids 152–53, 167 Einstein, Albert 30, 120, 175, 182, 228, 251, 273,
see also pi () cylindrical map projection 155 274, 275, 280, 304
346 INDEX

Einstein ring 209 Fermat’s little theorem 314, 317 Gaussian curve 192, 193
Eleatic school of philosophy 46, 47 Ferrari, Lodovico 105, 128 Gelfond, Alexander 197
Elements (Euclid) 21, 48, 50, 52–57, 66, 68, 82, Ferro, Scipione del 105, 128, 129, 130 general relativity 175, 274, 275, 280
87, 94, 120, 121, 154, 166, 229, 272, 308 Feuerbach, Karl Wilhelm 166 geodesic path 299, 299
common notions 55–56 Fewster, Rachel 290 geographic coordinates 151
postulates 55, 56–57, 56, 98, 103, 228 Fibonacci (Leonardo of Pisa) 27, 58, 64, 87, 88, geomagic squares 34
propositions 56, 57 91, 108, 134, 135 geometric series 113
ellipses 68, 69, 69, 104, 154, 226 Fibonacci ratios 110–11 geometry 14, 16, 20, 95
elliptic curves 226, 314, 320, 321, 321, 322 Fibonacci sequence 87, 106–11, 120, 121–22, algebraic geometry 283, 323
elliptic functions 226–27 123, 160, 161, 161, 196, 290 analytic geometry 69, 117, 148, 149
encryption see cryptography fields 233 Cartesian coordinates 117, 144–51, 195, 258,
Enigma machines 316, 316 Fields Medal 82, 196, 276, 325 282
entropy 218, 291 finite simple groups 318–19 conic sections 55, 68–69, 82, 103, 104, 154,
equals sign 126, 127 Fior, Antonio 128, 129, 130 155, 226
equations fluxions 153, 173, 174 elliptic functions 226–27
algebraic resolution 200–01 Fontana, Niccolò 215 Euclid’s Elements 21, 48, 50, 52–57, 66, 68, 82,
cubic 78, 87, 91, 98, 99, 102–05, 117, 200, four-colour theorem 312–13 87, 94, 120, 121, 154, 166, 228, 229, 272, 308
207, 215 four-dimensional space 274–75, 324–25 four-dimensional spacetime 274–75
Diophantine 80–81 Fourier, Joseph 75, 188, 216–17, 217 fractals 296, 306–11
indeterminate equations 97 Fourier analysis 216–17 hyperbolic geometry 54, 69, 69, 212, 228, 229
linear 77, 78, 94, 96, 102 Fourier trigonometry series 75 maxima and minima 142–43, 173, 174
polynomial 80, 131, 190, 200, 201, 206–207, fractal analysis 265, 308 non-Euclidean geometries 54, 57, 151, 212,
215, 230, 233 fractal-generating computer programs 311 228–29, 274
quadratic 28–31, 77, 78, 81, 87, 91, 96, 97, fractals 296, 306–11 pi () 33, 41, 45, 60–65, 69, 83
102, 200, 207 fractions 32, 44, 134 plane geometry 54
quartic 105, 117, 200, 201, 233 decimal fractions 135, 137 Platonic solids 21, 38, 41, 48–49, 54–55, 121,
quintic 200, 201 notation system 135 308
roots of an equation 206–07, 207, 208 Franklin, Benjamin 35 Poincaré conjecture 256, 258–59, 259, 265,
simultaneous 29, 239 Franklin, Philip 313 274, 324–25
structure of 30–31 fraud detection 290 polar coordinates 150, 150, 151
equilateral triangles 56, 56, 166, 166, 231–32, 232 Fréchet, Maurice 259 projective geometry 116, 154–55
Eratosthenes 67, 67, 124, 286–87 Frederick the Great of Prussia 191 spherical geometry 57
Eratosthenes’ sieve 66–67, 286–87 Freedman, Michael 324 synthetic geometry 148
Erdős, Paul 260, 293, 304 Frege, Gottlob 17, 50, 246, 273 tautochrone curve 167, 302
Erdős number 293, 304 Frénicle de Bessy, Bernard 276 topology 16, 212, 248, 256–59, 264, 283
Erdős-Bacon-Sabbath number 293 frequency analysis 314, 316 triangle theorem 166
Escher, M. C. 229, 249 Frey, Gerhard 321–22 trigonometry 16, 70–75, 116, 140, 226
Euclid 16, 38, 54, 69, 95, 112, 124, 260, 274, 286 Friedman, Harvey 272 23 unsolved problems 264, 266–67
see also Elements Frisius, Gemma 75 see also applied geometry
Eudoxus of Cnidus 44, 54, 73, 171 Fuller, R. Buckminster 258 Gersonides (Levi ben Gershon) 236
Euler circles 254 functoriality 303 Girard, Albert 206–07
Euler–Lagrange equation 175 fundamental theorem of algebra (FTA) 200, 201, Gödel, Kurt 267, 273, 286
Euler, Leonhard 34, 35, 48, 49, 65, 67, 75, 96, 124, 204–09, 215 Goldbach, Christian 181, 196
128, 131, 141, 170, 174, 175, 180–181, 181, 188, fuzzy logic 265, 300–01 Goldbach conjecture 181, 196
188–89, 188, 194–95, 196, 197, 200, 201, 207, fuzzy set theory 300 golden ratio 110, 111, 116, 118–23
215, 221, 236, 250, 256, 257–58, 276, 320, 321 Fyodorov, Evgraf 305 golden spiral 111, 122, 123
Eulerian paths 195 Goldie, Alfred 280
Euler’s identity 75, 181, 197 Gonthier, Georges 312

G
Euler’s number 174, 180, 186–91, 194, 197 Gorenstein, Daniel 318, 319
Euler’s polyhedral formula 48, 49, 257–58, 257 Govert de Bruijn, Nicolaas 305
exponential function 141, 174, 190 graph theory 99, 194–95
exponential growth 112–13, 141, 190, 195 Cayley graph 319, 319
Eye of Horus 32 Galileo Galilei 75, 153, 162, 166, 182, 216, 298 coordinates and axes 104, 147–48, 149, 149, 195
Galois, Évariste 201, 213, 230, 231, 231, 233, 318 matrices 241
Galois groups 233, 303 rhumb lines 125

F
Galois theory 231, 233 weighted graphs 195
Galton, Francis 193, 268, 269, 270 Graunt, John 220
Galton board 193, 269 gravitational lensing 209
gambler’s fallacy 185 gravity 182, 183, 228, 274, 298
false position method 29, 32–33 games of chance 117, 159, 163–64, 165, 184, greater than symbol 127
Fast Fourier Transform (FFT) 216 185, 192, 271 Greek mathematics 20–21, 63, 73, 78, 89, 95,
Fatou, George 310 Gates, Bill 91 103, 152, 200
Fauvel, John 55 Gaudí, Antoni 311 see also specific mathematicians
Fermat, Pierre de 40, 67, 69, 80, 81, 117, 142, Gauss, Carl Friedrich 54, 66, 67, 126, 127, 181, Gregory, James 62, 65, 171, 173
150, 153, 162, 163, 165, 170, 173, 184, 196, 314 193, 199, 200, 206, 208–09, 208, 212, 215, Griggs, Jerrold 254
Fermat’s last theorem 117, 227, 236, 283, 302, 226, 228, 229, 238, 239, 240, 251, 268, 269, group theory 200, 201, 213, 230–33, 240
303, 320–23 280, 302 simple groups 318–19
 INDEX 347

Guillaud, Jean 65 imaginary numbers 30, 75, 117, 128–31, 151, Klein bottle 248, 258
Gurevitch, Michael 292–93 197, 207, 214, 215 Knauf, Andreas 124
Guthrie, Francis 312–13 see also complex numbers Kneser, Hellmuth 206
incompleteness theorem 273, 286 Koch, Helge von 250, 309
Indian mathematics 21, 27, 34–35, 81, 86, 90 Koch curve/snowflake 309, 310

H see also specific mathematicians

indices 99
inductive reasoning 21
Kochen, Manfred 292
Kovalevskaya, Sofia 82
Kummer, Ernst 280
infinite monkey theorem 252, 278–79 Kurzweil, Ray 113
Hadamard, Jacques 131, 260–61, 261 infinite-dimensional toplogy 259
Haken, Wolfgang 313 infinitesimal calculus 47, 126

L
Halayudha 158 infinitesimals 47, 142, 143, 170, 171, 174, 175
Haldane, J. B. S. 98 infinity 21, 91, 170, 252, 267, 272, 278
Halley, Edmond 182 information theory 289, 291
Hamilton, Richard 325 integers 41, 44, 66, 80–81, 124, 131, 196, 280, 304
Hamilton, William 146, 212, 214, 234–35, 235, 313 ring of integers 280 Lagrange, Joseph-Louis 167, 175, 181, 201, 201,
Hardy, G. H. 26, 276, 277 see also number theory 280
Harmonia 42 integration 170, 172, 173, 174, 175 Lambert, Johann Heinrich 45
harmonic analysis 302, 303 internet security 67, 317 Landau, Edmund 260
harmonic polynomials 209 invariant theory 280 Landauer, Rolf 291
harmonic spectrum 217 inverse probability 198, 199 Langlands, Robert 302–03, 303
Harriot, Thomas 76, 127, 176, 177 irrational numbers 15, 21, 28, 30, 41, 44–45, 54, Langlands Program 302–03
Harrison, John 167 62, 96, 97, 120, 131, 137 Laplace, Pierre-Simon 141, 162, 163, 165, 175,
Hartley, Ralph 291 Islamic mathematics 16, 35, 45, 72, 86–87, 95–99 193, 195, 198, 208, 212, 218–19, 219, 268, 296
al-Hasib 74 see also specific mathematicians Laplace’s demon 218–19
Hausdorff, Felix 309 isosceles triangles 57 The Last Supper (Leonardo da Vinci) 122, 122
Heath, Sir Thomas L. 63, 72 Latin squares 35
Heawood, Percy 312, 313 lava lamps 317, 317

J
Heesch, Heinrich 313 law of large numbers (LLN) 165, 180, 184–85
Henderson, David W. 229, 254 law of single chance 279
heptadecagon 208 laws of motion 182–83, 218
Hermite, Charles 188, 190 least squares method 269
Heron of Alexandria 166 Jacobi, Carl 213, 227, 227 Lebesgue, Henri 137
Hewlett Packard 58 Jacobson, Nathan 131 Legendre, Adrien-Marie 66, 227, 260
hexagons 63, 68 Jacquard, Joseph-Marie 222, 224 Leibniz, Gottfried 50, 62, 65, 116, 117, 142, 143,
hexagrams 34, 155, 177 Janko, Swonimir 318 174, 175, 177, 185, 190, 207, 223, 244, 254, 291
hexahedra 48 Japanese mathematics 26–27 Leonardo da Vinci 64, 120, 121, 122, 122
hieroglyphs 33 Jefferson, Thomas 137 less than symbol 127
Higgs boson 233 Jia Xian 158, 160 L’Huilier, Simone 258
Hilbert, David 81, 229, 250, 259, 264, 266–67, Jones, William 62 Lie, Sophus 233
267, 272, 280, 286 Jordan, Camille 318 Lie groups 233
Hill, Ted 290 Jordan–Hölder theorem 318 linear equations 77, 78, 94, 96, 102
Hindu–Arabic numeration 24, 26, 27, 45, 58, 86, Julia, Gaston 310 linear geometric transformations 240, 241
87, 91, 94, 96, 116 Julia set 310, 311 linear perspective 154, 154, 155
Hinton, Charles 234, 274 Liouville, Joseph 230, 237, 252
Hipparchus 73, 73 Listing, Johann 248, 256, 258

K
Hippasus 41, 44, 45, 45, 121, 206 Liu Hui 21, 64, 83
Hippocrates of Chios 54, 55 Livio, Mario 122
Hire, Philippe de la 154 Llull, Ramon 254
Hobson, E. W. 221 Lobachevsky, Nicolai Ivanovich 14, 54, 212, 229
Hofstadter, Douglas 131 Kanigel, Robert 62 logarithmic scales 140, 141
Hokusai, Katsushika 311, 311 al-Karaji 98, 100–01 logarithms 58, 75, 99, 116, 138–41, 188
Hölder, Otto 318 Karinthy, Frigyes 292, 293, 304 logic 16, 21
Hooke, Robert 182, 183 Katahiro, Takebe 83 Boolean logic 176, 213, 242–47, 286, 291, 300,
Huygens, Christiaan 152, 162, 165, 167, 190 Katz, Nick 322, 323 301
Hypatia of Alexandria 81, 82, 82 Kempe, Alfred 313 fuzzy logic 265, 300–01
hyperbolic geometry 54, 69, 69, 212, 228, 229 Kepler, Johannes 48, 49, 108, 120, 122, 141, Russell’s paradox 272–73
142–43, 143, 172, 182, 221, 227, 298, 305 syllogistic logic 50–51, 244
Khavinson, Dmitry 209 23 unsolved problems 264, 266–67

I
Khayyam, Omar 68, 98, 99, 102–05, 105, 129, Zeno’s paradoxes 46–47, 117, 170–71
158, 160 logic gates 247, 247
al-Khwarizmi 30, 64, 78, 87, 91, 94, 94, 95–96, longitude problem 167
99, 100 Lorentz, Henrik 275
I Ching (Book of Changes) 34, 177, 177 Killian, Charles 254 Lorenz, Edward 218, 265, 296–98, 296
icosahedra 48, 49, 49 al-Kindi 134, 314, 316 Lorenz attractor 218, 296, 297, 297
icosian game 235 Kirchhoff’s laws of voltage and current 241 Lovelace, Ada 213, 223, 224–25, 225
ideal numbers 280 Klein, Felix 229, 230, 248, 258 loxodromes 125, 125
348 INDEX

Lucas, Édouard 108, 124, 255 music and mathematics 42–43, 111 Langlands Program 302–03
Ludlow, Peter 316 musical scales 42, 43 magic squares 34–35
Łukasiewiecz, Jan 300 matrices 238–41
Mersenne primes 124, 161

M N Pascal’s triangle 100, 158–61

pi () 33, 41, 45, 60–65, 69, 83
prime number theorem 260–61
Riemann hypothesis 213, 250–51, 261, 267
Nadi Yali yantra 90 six degrees of separation 292–93, 304
Maclaurin, Colin 174 naive set theory 264, 273 social mathematics 304
Madhava of Kerala 138 Napier, John 74, 75, 116, 139–41, 139, 177, 188, 222 statistics 269–71
magic squares 34–35 Napier’s Bones 139 taxicab numbers 276–77
Malthus, Thomas 112, 190 natural logarithms 141, 188 Tower of Hanoi 255
Mandelbrot, Benoit 265, 296, 308, 309, 309, 310 natural numbers 124, 131, 213, 236 transfinite numbers 252–53
Mandelbrot set 308, 309, 310–11, 311 Nave, Annibale della 129 zero 27, 86, 88–91, 105, 240, 250, 251
Manhattan Project 202 negative numbers 26, 28, 30, 69, 76–79, 89, 91, Nunes, Pedro 125
Markowsky, George 123 104, 105, 129, 130, 200, 206, 207, 214, 215
Marquardt, Stephen 123 Neumann, Genevra 209

O
Masjid-i Kabud (Blue Mosque) 105 Neumann, John von 288, 289
mathematical induction 98 New Mathematics movement 282
Matiyasevich, Yuri 81 Newcomb, Simon 290
matrices 238–41 Newman, Max 222
matrix mechanics 241 Newton, Isaac 83, 100, 117, 126, 142, 143, 171, octagon 63
maxima and minima 142–43, 173, 174 173–74, 182–83, 183, 185, 199, 218, 227, 298 octahedra 48, 49, 49
Maxwell, James Clerk 275 Newtonian calculus 100, 117, 126, 143, 171, Ohm, Martin 120
Mayan mathematics 26, 27, 89, 238–39 173–74, 185, 199, 218, 298 Olmecs 59
Mayer, Jean 313 Newton’s laws of motion 182–83, 218 Ore, Oystein 313
Melville, Herman 167 Noether, Emmy 233, 264, 280–81, 281 Oresme, Nicole 146, 150, 171–72
Menabrea, Luigi 225 Noether’s theorem 280 orrery 218
Menelaus of Alexandria 57 non-Euclidean geometries 54, 57, 151, 212, Ostrowski, Alexander 206, 209
Mengoli, Pietro 141 228–29, 274 Oughtred, William 58, 116, 127, 138
mensuration 33 non-orientable surfaces 249 Oxford Calculators 172
Mercator, Gerardus 125, 155 normal distribution 181, 188, 192–93, 199
Mercator, Nicholas 138, 141 number systems
Méré, Antoine Gombaud, Chevalier de 159, 163–64
Merilees, Philip 296, 297
Mersenne, Marin 67, 117, 124, 153, 164–65
abacus 26–27, 58–59, 59, 87
decimal system 24, 26, 27, 32, 89, 96, 132–37
Hindu–Arabic 24, 26, 27, 45, 58, 86, 87, 91, 94,
P
Mersenne primes 124, 161, 255 96, 116
metempsychosis 41–42 irrational numbers 15, 21, 28, 30, 41, 44–45, P versus NP problem 286
method of exhaustion 170, 171 54, 62, 96, 97, 120, 131, 137 Pacioli, Luca 35, 116, 120, 121, 121, 158, 164
metric space 259 logarithms 58, 75, 99, 116, 138–41, 188 parabolas 31, 31, 69, 69, 104, 104, 153, 154, 190
metric system 134, 137 negative numbers 26, 28, 30, 69, 76–79, 89, parabolic objects 31, 31
Mihăilescu, Preda 236, 237 91, 104, 105, 129, 130, 200, 206, 207, 214, 215 parallel postulate (PP) 103, 212, 228–29, 228
Milgram, Stanley 292–93, 304 notation 126–27 Parmenides 46, 47
Minkowski, Hermann 264, 274, 275, 275 positional numbers 22–27, 58, 59 Parthenon 120
Mirzakhani, Maryam 82 quaternions 146, 151, 212, 214, 234–35 Pascal, Blaise 68, 117, 153, 155, 156, 158–61,
Möbius, August 194, 212, 248–49, 249 rod numerals 26, 76, 77, 77 159, 162, 164, 165, 167, 184, 222, 223
Möbius function 249 number theory Pascal’s triangle 100, 156–61
Möbius inversion formula 249 Benford’s law 290 Peano, Giuseppe 272, 273
Möbius net 249 binary numbers 27, 176–77, 265, 289, 291, 300 Pearson, Karl 192, 268, 269, 270–71, 271
Möbius plane 249 binomial theorem 87, 100–01 Peirce, Charles Sanders 244, 246
Möbius strip 212, 248–49, 258 Bourbaki group 282–83 pendulums 152, 167, 221, 298, 299
Möbius transformations 249 Catalan’s conjecture 236–37 Penrose, Roger 123, 305
modular (“clock”) arithmetic 303 complex numbers see complex numbers Penrose tiling 305
modular forms theory 276 elliptic functions 226–27 pentagons 41, 44, 63
modularity theorem 320, 321, 322 Eratosthenes’ sieve 66–67, 286–87 pentagrams 41, 121
Moivre, Abraham de 75, 162, 181, 184, 188, Euler’s identity 75, 181, 197 Percy, David 197
192–93, 193, 197, 198–99, 202, 220 Euler’s number 174, 180, 186–91, 194, 197 Perelman, Grigori 256, 265, 324, 325
Monte Carlo methods 202, 203 exponential growth 112–13, 141, 190, 195 perfect numbers 40–41
Moore, Gordon 112 Fermat’s last theorem 117, 227, 236, 283, 302, permutation groups 230, 233
Morse Code 177 303, 320–23 permutations 191, 201
Moscow papyrus 32, 33, 62 Fibonacci sequence 87, 106–11, 120, 121–22, perspective 116, 154–55, 154
motion, paradoxes of 46–47 123, 160, 161, 161, 196, 290 Phidias 120
multiplication 14, 137, 138, 139 finite simple groups 318–19 philosophy and mathematics 15, 17, 41, 49, 50,
matrix multiplication 239–40, 240, 241 Goldbach conjecture 181, 196 218, 244
symbol 126, 127 graph theory 99, 194–95, 241 pi () 33, 41, 45, 60–65, 69, 83
zero and 91 group theory 200, 201, 213, 230–33, 240 piano keyboard 111
 INDEX 349

Pingala 108, 111 Pythagoras’s theorem 38–39, 39, 40, 44, 63, 72, Robertson, Neil 312
pitch and frequency 216–17, 216 149, 320–21 Roberval, Gilles Personne de 152, 153
place value see positional numbers Pythagorean community 21, 40, 41, 43, 82, 121 robots 259, 301
plaintext 315, 316, 317 Pythagorean triples 38, 38, 39, 72, 111, 321 rod numerals 26, 76, 77, 77
plane geometry 54 roller-coasters 149, 249
planets Roman mathematics 21, 59, 134
laws of motion 141, 142, 172, 172, 182–83, 221
planetary orbits 49, 65, 142, 172, 221, 227, 257
Plato 21, 38, 41–42, 48, 49, 49, 55, 56, 121
Q Roman numerals 26, 87, 134
Roomen, Adriaan van (Romanus) 64–65
roulette wheels 164
Platonic solids 21, 38, 41, 48–49, 54–55, 121, 308 RSA algorithm 314–15, 317
Plimpton 322 tablet 38, 72 al-Qalasadi 97 “rubber-sheet geometry” see topology
plus and minus symbols 30, 79, 126, 127 quadratic equations 28–31, 77, 78, 81, 87, 91, Rubik’s cube 232
Poincaré, Henri 194, 229, 257, 257, 259, 264, 96, 97, 102, 200, 207 Ruffini, Paolo 102, 200, 201, 230
265, 282, 283, 296, 298–299, 302, 309, 310 completing the square 30, 96, 97 Russell, Bertrand 17, 46, 252, 264, 272–73, 273
Poincaré conjecture 256, 258–59, 265, 274, 324–25 quadratic formula 30 Russell’s paradox 272–73
Poisson, Siméon 185, 213, 220, 220 quadratic function 31, 31 Rutherford, William 65
Poisson distribution 213, 220 quadratic reciprocity theorem 302
polar coordinates 150, 150, 151 quantum computing 289

S
polyalphabetic cipher 316 quantum mechanics 16, 75, 131, 175, 227, 235
polygonal numbers 40 quantum physics 219, 241
polygons 42, 48, 63, 63, 64, 83, 170, 171, 232, quantum theory 251
237, 305 quartic equations 105, 117, 200, 201, 233
construction 208 quaternions 146, 151, 212, 214, 234–35 Saccheri, Giovanni 228, 229
polyhedra 38, 41, 48–49, 266–67 Quetelet, Adolphe 268, 269 Salamis Tablet 58
Euler’s polyhedral formula 48, 49, 257–58, 257 Quetelet Index (BMI) 269 Sallows, Lee 34
polynomial equations 80, 131, 190, 200, 201, quincunx 193, 269 al-Samaw’al 98–99
206–07, 215, 230, 233 quintic equations 200, 201 Sanders, Daniel P. 312
polynomials 98, 99, 101, 181, 206, 207, 208, 209 Sarnak, Peter 323
Poncelet, Jean-Victor 154, 155 Sauveur, Joseph 216
positional numbers 22–27, 58
base-2 (binary) system 176
base-10 (decimal) system 20, 24, 24–25, 27,
R Savage, Carla 254
Schooten, Frans van 150
Schröder, Ernst 246–47
116, 176 Schrödinger’s wave equation 175
base-20 (vigesimal) system 26, 27, 59 Rahn, Johann 127 Scientific Revolution 87, 116, 180
base-60 (sexagesimal) system 24, 25–26, 26, 73 Ramanujan, Srinivasa 276, 277, 277 search engines 241, 247
zero 27, 86, 88–91, 88–91, 105, 240 random variation 193 Selberg, Atle 260
see also binary system; decimal system rational numbers 44, 45, 80–81, 96, 131 self-similarity 309, 310, 310, 311, 311
Possel, René de 283 real numbers 45, 105, 128, 129, 131, 141, 151, set theory 45, 213, 252, 253, 254, 267, 282, 283,
powers 127, 139, 140, 197, 215, 236 190, 197, 207, 209, 214 300
Price, Richard 198 Recorde, Robert 126, 127 fuzzy set theory 300
prime numbers 15, 66, 117, 196, 277, 302 recreational mathematics naive set theory 264, 273
Eratosthenes’ sieve 66–67 magic squares 34–35 seven bridges of Königsberg 180, 194–95, 195
Mersenne primes 124, 161, 255 Tower of Hanoi 255 Seymour, Paul 312
prime number theorem 260–61 recurring decimals 137 Shamir, Adi 314
Riemann hypothesis 213, 250–51, 261, 267 Regiomontanus (Johannes Müller von Shanks, Daniel 65
probability theory 117, 162–65, 180, 181, 219, 291 Königsberg) 75 Shannon, Claude 176, 244, 247, 286, 289, 291, 291
Bayes’ theorem 198–99 Regius, Hudalrichus 124 Shechtman, Dan 305
Buffon’s needle experiment 202–03 regression to the mean 270, 270, 271 Sheil-Small, Terrence 209
central limit theorem 184 relative frequency 163, 164 Shimura, Goro 320, 321
chaos theory 16, 257, 265, 283, 294–99, 310 relativity theories 175, 182, 251, 264, 274, 275, 280 Shishikura, Mitsuhiro 311
infinite monkey theorem 278–79 Renaissance 87, 116–17, 120, 122 Sierpinski, Waclaw 158, 161
inverse probability 198, 199 reverse mathematics 272 Sierpinski triangles 158, 160, 161, 309, 311
law of large numbers 165, 180, 184–85 Rhind papyrus 32–33, 33, 62, 72, 73, 83, 94, 170 simple groups
law of single chance 279 rhumb lines 125 finite simple groups 318–19
normal distribution 181, 188, 192–93, 199 Ribet, Ken 322 The Monster 319
Pascal’s triangle 100, 156–61 Ricci flow 325 simultaneous equations 29, 239
Poisson distribution 213, 220 Riemann, Bernhard 66, 175, 212–13, 214, 228, sine, cosine, and tangent 74, 75
Proclus 54, 55, 228–29 229, 250–51, 251, 258, 260, 261 sine waves 216, 217, 302
projectile flight 31, 31, 75 Riemann hypothesis 213, 250–51, 261, 267 singularities 113, 325
projective geometry 116, 154–55 Riemann zeta function 131, 213, 251 Sissa ben Dahir 112–13
proof by exhaustion 313 right-angled triangles 21, 38–39, 44, 63, 72, 75, six degrees of separation 292–93, 304
Ptolemy 64, 73, 82, 89, 125 111, 138 slide rule 58, 116, 138, 141
public key cryptography 67, 227, 233, 317, 317 see also Pythagoras’s theorem; Pythagorean Smale, Stephen 267, 324
pure mathematics 15 triples; trigonometry small world theory 292–93
pyramids 33, 62, 65, 73, 121 rings 233, 280, 281 Smith, Henry 308
Pythagoras 15, 21, 36–43, 41, 43, 44, 48, 50, 59, Rivest, Ron 314 Smith, Stephen D. 318
81, 216 Robert of Chester 87 Snell, Willebrord 125
350 INDEX

social mathematics 304 terminating decimals 137, 190 Viète, François 64, 75, 80, 94, 99, 105, 127, 146,
social networks 195, 292–93, 304 tessellations 305, 305 150, 172
Socrates 47, 49, 50 tesseract 234, 274 Vinogradov, Ivan 196
Somayagi, Nilakantha 83 tetrahedra 48–49, 49, 160, 161 Virahanka 108
soroban 26–27, 27, 58, 59 Thales of Miletus 32, 38, 39, 40, 54, 55 Virgil 111
Soul conjecture 325 Theaetetus 48, 54 virtual reality games 235
spacetime 151, 264, 274, 275 Theodorus of Cyrene 44 Vitruvian Man 122–23
special relativity 175, 275 Theodosius of Bithynia 57 Vitruvius 123
speech recognition software 216, 217 thermodynamics 217, 218, 219 Viviani, Vincenzo 166
speed of light 274–75 Thiele, Rüdiger 267 Vogel, Kurt 81
spherical geometry 57 Thomas, Robin 312 Voltaire 24
spherical trigonometry 74, 74 Thompson, John G. 319
square matrices 239, 240, 241 three-body problem 221, 296, 298–99

W
squaring the circle 63–64, 64 three-dimensional space 102, 147, 148, 234–35,
stadium paradox 46, 47 249, 258, 259, 274, 297, 324
standard deviation 270–71 Thurston, William 229
statistical mechanics 218, 219 Tijdeman, Robert 236, 237
statistics 268–71 topology 16, 212, 248, 256–59, 264, 283 Waerden, Bartel Leendert Van der 280
chi-squared test 271 algebraic topology 259, 282 Wallis, John 79, 91, 226, 227
law of large numbers 165, 180, 184–85 four-colour theorem 312–13 Wang Fau 64
normal distribution 181, 188, 192–93, 199 fractals 296, 306–11 Wang Xiaotong 102
Poisson distribution 213, 220 graph theory 99, 194–95, 241 Watson, George 221
Venn diagrams 244, 246, 246, 254 infinite-dimensional topology 259 Watts, Duncan J. 292, 293
Steiner, Jakob 166 Poincaré conjecture 256, 258–59, 259, 265, Watts–Strogatz random graph model 292
Stemple, Joel 313 274, 324–25 wave analysis 217, 221, 302
Stevin, Simon 116, 134, 134, 135, 137, 172 Torricelli, Evangelista 173 Weierstrass, Karl 209, 226, 227, 308
Stewart, Ian 245 Tower of Hanoi 255 Weierstrass function 308, 310
stochastic matrices 241 transcendental numbers 62, 190, 252–53, 253 Weil, André 264, 283, 283, 302
stock markets 192, 279, 301 transfinite numbers 252–53 Wessel, Casper 215
string vibrations 216–17 transformation matrices 241 wheat on a chessboard problem 112–13
Strogatz, Steven 292, 293, 299 travelling salesperson problem 195 Whitehead, Alfred North 69, 134, 273
Struik, Dirk 151 Treviso Arithmetic 91, 91, 126 Whitehead, Henry 324
suanpan 59, 59 triangles 57, 65 Widman, Johannes 126, 127
subtraction 14 equilateral 56, 56, 166, 166, 231–32, 232 Wilder, Raymond Louis 259
symbol 79, 126, 127 isosceles 57
Wiles, Andrew 81, 282, 283, 302, 303, 321,
Sudoku puzzles 34, 35 right-angled 21, 38–39, 44, 63
322–23, 323
Sumerians 20, 24, 88 Sierpinski triangles 158, 160, 161, 309, 311
Wilmshurst, Alan 209
summation 170 triangle postulate 38
Wolpert, David 218
Sun Hong Rhie 209 triangle theorem 166
women mathematicians 82
Sun Tzu 77 see also trigonometry
superscripts 127 see also specific mathematicians
triangulation stations (trig points) 75
surds 30 World Wide Web 265
trigonometry 16, 70–75, 116, 140, 226
syllogistic logic 50–51, 244 planar trigonometry 74 worldline 275, 275
Sylvester, James Joseph 239, 239 spherical trigonometry 74, 74 Wren, Christopher 153
symbolic algebra 21, 94, 99 trigonometric tables 72, 73, 104 Wrench, John 65
symmetric encryption 315, 316–17 trigrams 34 Wright, Edward 277
symmetries 231–32, 233, 241, 280, 319 Trueb, Peter 62, 65
synthetic geometry 148 Tukey, John 216
Turing, Alan 265, 272, 287–89, 287, 316
Turing machine 284–89 Y,Z
T Turing test 265, 289
al-Tusi 99, 229
type theory 273 Yang Hui 160
Z3 289
Typex 316
Taimina, Daina 229 Zadeh, Lotfi 265, 300
Takakaze, Seki 240 Zagier, Don 261

U,V
tally marks 14, 20 Zeising, Adolf 123
Ta(n) 24, 276, 277 Zeno of Elea 21, 46–47, 47, 252
Taniyama, Yutaka 320, 321 Zeno’s paradoxes 46–47, 117, 170–71
Taniyama–Shimura conjecture 320, 321, 322 Zermelo, Ernst 272
Tao, Terence 196 Ulam, Stanislaw 203 zero 27, 86, 88–91, 105, 240
Tarski, Alfred 300 vanishing points 154, 155 non-trivial zeros 250, 251
Tartaglia (Niccolò Fontana) 105, 128, 129–30, Varian, Hal 290 Zhao, Ke 237
158, 164, 215 Varignon, Pierre 122 Zhu Shijie 160
tautochrone curve 167, 302 vectors 151 Zu Chongzhi 62, 64, 83
taxicab numbers 276–77 Venn, John 244, 246, 254 Zu’s ratio 83
Taylor, Richard 323 Venn diagrams 244, 246, 246, 254 Zuse, Konrad 225, 289
 351

QUOTATIONS
The following primary quotations are attributed
to people who are not the key figure for the THE ENLIGHTENMENT MODERN MATHEMATICS
relevant topic.
186 One of those strange numbers that 268 Statistics is the grammar of science
are creatures of their own Karl Pearson, British mathematician
ANCIENT AND CLASSICAL PERIODS Ian Stewart, British mathematician and statistician
197 The most beautiful equation 276 Rather a dull number
60 Exploring pi is like exploring
Keith Devlin, British mathematician G. H. Hardy, English mathematician
the Universe
David Chudnovsky, Ukrainian–American 198 No theory is perfect 278 A million monkeys banging
mathematician Nate Silver, American statistician on a million typewriters
Robert Wilensky, American
70 The art of measuring triangles 200 Simply a question of algebra computer scientist
Samuel Johnson, English writer Robert Simpson Woodward, American
engineer, physicist, and mathematician 280 She changed the face of algebra
80 The very flower of arithmetic
Hermann Weyl, German mathematician
Regiomontanus, German mathematician 204 Algebra often gives more than is
and astronomer asked of her 291 A blueprint for the digital age
Jean d’Alembert, French mathematician Robert Gallagher, American engineer
82 An incomparable star in the
and philosopher
firmament of wisdom 294 A small positive vibration can
Martin Cohen, British philosopher change the entire cosmos
Amit Ray, Indian author
THE 19TH CENTURY
302 A grand unifying theory of
THE MIDDLE AGES 218 The imp that knows the positions of mathematics
every particle in the Universe Edward Frenkel, Russian–American
92 Algebra is a scientific art
Steven Pinker, Canadian psychologist mathematician
Omar Khayyam, Persian mathematician
and poet 221 An indispensable tool in applied 306 Endless variety and unlimited
mathematics complication
106 The ubiquitous music of the spheres
Walter Fricke, German astronomer and Roger Penrose, British mathematician
Guy Murchie, American writer
mathematician
318 Jewels strung on an as-yet
112 The power of doubling
226 A new kind of function invisible thread
Ibn Khallikan, Islamic scholar and biographer
W. W. Rouse Ball, British mathematician Ronald Solomon, American mathematician
and lawyer
320 A truly marvellous proof
THE RENAISSANCE 234 Just like a pocket map Pierre de Fermat, French lawyer
and mathematician
attributed to Peter Tait, British
118 The geometry of art and life physicist and mathematician, by
Matila Ghyka, Romanian novelist Silvanus Phillips Thompson, British
and mathematician physicist and engineer
124 Like a large diamond 238 The matrix is everywhere
Chris Caldwell, American mathematician from the film The Matrix
152 A device of marvellous invention 250 The music of the primes
Evangelista Torricelli, Italian physicist Marcus du Sautoy, British mathematician
and mathematician and author
162 Chance is bridled and governed by law 252 Some infinities are bigger than others
Boëthius, Roman senator John Green, American author
168 With calculus I can predict the future 260 Lost in that silent, measured space
Steven Strogatz, American mathematician Paolo Giordano, Italian author
ACKNOWLEDGEMENTS
Dorling Kindersley would like to thank Gadi Farfour, Library: Science Source (b). 134 iStockphoto.com: Library: Dr Mitsuo Ohtsuki (cb). 253 Alamy Stock
Meenal Goel, Debjyoti Mukherjee, Sonali Rawat, sigurcamp (bl). 136 Alamy Stock Photo: George Oze (br). Photo: INTERFOTO (bl). 255 Dreamstime.com: Antonio
and Garima Agarwal for design assistance; Rose 137 Alamy Stock Photo: Hemis (tl). 139 Alamy Stock De Azevedo Negrão (cr). 257 Alamy Stock Photo:
Blackett-Ord, Daniel Byrne, Kathryn Hennessy, Mark Photo: Classic Image (tr). 140 Science Photo Library: Science History Images (tr). 259 Alamy Stock Photo:
Silas, and Shreya Iyengar for editorial assistance; and Oona Stern (clb). 141 Alamy Stock Photo: Pictorial Press Wenn Rights Ltd (tr). 261 Getty Images: ullstein bild Dtl.
Gillian Reid, Amy Knight, Jacqueline Street-Elkayam, Ltd (tl). Getty Images: Keystone Features / Stringer / (tr). 267 Alamy Stock Photo: History and Art Collection
and Anita Yadav for production assistance. Hulton Archive (bc). 143 Wellcome Collection http:// (tr). 270 Alamy Stock Photo: Chronicle (tl). 271 Science
creativecommons.org/licenses/by/4.0/: (tr). 146 Photo Library: (bl). 273 Getty Images: John Pratt /

PICTURE CREDITS
Alamy Stock Photo: IanDagnall Computing (bl). 147 Stringer / Hulton Archive (cla). 274 NASA: NASA’s
Science Photo Library: Royal Astronomical Society Goddard Space Flight Center (br). 275 Getty Images:
(br). 150 Getty Images: Etienne DE MALGLAIVE / Keystone / Stringer / Hulton Archive (tr). 277 Alamy
The publisher would like to thank the following for their Gamma-Rapho (tr). 159 Rijksmuseum, Amsterdam: Stock Photo: Granger Historical Picture Archive (tr). 279
kind permission to reproduce their photographs: (bl). 160 Alamy Stock Photo: James Nesterwitz (clb). Getty Images: Eduardo Munoz Alvarez / Stringer / Getty
Gwen Fisher: Bat Country was selected as a 2013 Images News (cra); ullstein bild Dtl. (bl). 281 Alamy Stock
(Key: a-above; b-below/bottom; c-centre; f-far; l-left; Burning Man Honorarium Art Project. (tl). 163 123RF.com: Photo: FLHC 61 (cra). 283 Bridgeman Images: Private
r-right; t-top) Antonio Abrignani (tr). Alamy Stock Photo: Chris Collection / Archives Charmet (cla). 287 Alamy Stock
Pearsall (clb, cb, cb/blue). 164 Alamy Stock Photo: i Photo: Granger Historical Picture Archive (tr). Getty
25 Getty Images: Universal History Archive / Universal creative (br). 171 Alamy Stock Photo: Stefano Ravera Images: Bletchley Park Trust / SSPL (bl). 289 Alamy
Images Group (crb). Science Photo Library: New York (ca). 172 Getty Images: DE AGOSTINI PICTURE Stock Photo: Ian Dagnall (tl). 291 Alamy Stock Photo:
Public Library (bl). 27 Alamy Stock Photo: Artokoloro LIBRARY (bl). 173 Alamy Stock Photo: Granger Science History Images (cr). 296 Alamy Stock Photo:
Quint Lox Limited (tr); NMUIM (clb). 29 Alamy Stock Historical Picture Archive (br). 174 Alamy Stock Photo: Jessica Moore (tr). Science Photo Library: Emilio Segre
Photo: Historic Images (cra). 31 SuperStock: Stocktrek World History Archive (tc). 175 Alamy Stock Photo: Visual Archives / American Institute of Physics (bl). 297
Images (crb). 32 Getty Images: Werner Forman / INTERFOTO (tr). 177 Alamy Stock Photo: Chronicle Alamy Stock Photo: Science Photo Library (br). 301
Universal Images Group (cb). 33 Getty Images: DEA (crb). 182 Getty Images: Science Photo Library (bc). Alamy Stock Photo: Aflo Co. Ltd. (clb). 303 Dorling
PICTURE LIBRARY / De Agostini (crb). 35 Getty Images: 183 Library of Congress, Washington, D.C.: Kindersley: H. Samuel Ltd (cra). Institute for Advanced
Print Collector / Hulton Archive (cla). 38 Alamy Stock LC-USZ62-10191 (b&w film copy neg.) (bl). 185 Alamy Study: Randall Hagadorn (bl). 308 Getty Images:
Photo: World History Archive (tr). 41 Alamy Stock Stock Photo: Interfoto (tr). Getty Images: Xavier Laine / PASIEKA / Science Photo Library (tr). 309 Science Photo
Photo: Peter Horree (br). 42 Getty Images: DEA Picture Getty Images Sport (bc). 188 Wellcome Collection Library: Emilio Segre Visual Archives / American Institute
Library / De Agostini (br). 43 Alamy Stock Photo: World http://creativecommons.org/licenses/by/4.0/: (bl). Of Physics (tr). 310 Alamy Stock Photo: Steve Taylor
History Archive (tl). SuperStock: Album / Oronoz (tr). 190 Alamy Stock Photo: Peter Horree (clb). 191 Alamy ARPS (bc). 311 Getty Images: Fine Art / Corbis Historical
45 Alamy Stock Photo: The History Collection (tr). Stock Photo: James King-Holmes (tr). 193 Alamy Stock (clb). 313 Getty Images: f8 Imaging / Hulton Archive
47 Rijksmuseum, Amsterdam: Gift of J. de Jong Photo: Heritage Image Partnership Ltd (bl). 196 Getty (crb). 315 Dorling Kindersley: Royal Signals Museum,
Hanedoes, Amsterdam (bl). 49 Dreamstime.com: Images: Steve Jennings / Stringer / Getty Images Blandford Camp, Dorset (bc). 316 Alamy Stock Photo:
Vladimir Korostyshevskiy (bl). 51 Dreamstime.com: Entertainment (crb). 201 Alamy Stock Photo: Interfoto (bl). 317 Getty Images: Matt Cardy / Stringer /
Mohamed Osama (tr). 54 Wellcome Collection http:// Classic Image (bl). 203 Wellcome Collection http:// Getty Images News (clb). 323 Science Photo Library:
creativecommons.org/licenses/by/4.0/: (bl). 55 creativecommons.org/licenses/by/4.0/: (tr). 206 Frederic Woirgard / Look At Sciences (bl). 325 Avalon:
Science Photo Library: Royal Astronomical Society (cra). Alamy Stock Photo: Science History Images (tr). 208 Frances M. Roberts (tr)
59 Alamy Stock Photo: Science History Images (br). 63 Alamy Stock Photo: Science History Images (tr). Getty
Wellcome Collection http://creativecommons.org/ Images: Bettmann (tl). 209 Alamy Stock Photo: NASA All other images © Dorling Kindersley
licenses/by/4.0/: (bl). 65 Alamy Stock Photo: National Image Collection (clb). 217 Getty Images: AFP For further information see: www.dkimages.com
Geographic Image Collection (tl). NASA: Images courtesy Contributor (br). Wellcome Collection http://
of NASA / JPL-Caltech / Space Science Institute (crb). 67 creativecommons.org/licenses/by/4.0/: (cla). 218
Alamy Stock Photo: Ancient Art and Architecture (tr). Getty Images: Jamie Cooper / SSPL (bc). 219 Getty
73 Alamy Stock Photo: Science History Images (bl, cra). Images: Mondadori Portfolio / Hulton Fine Art Collection
75 Dreamstime.com: Gavin Haskell (tr). 77 Getty (tr). 220 Alamy Stock Photo: The Picture Art Collection
Images: Steve Gettle / Minden Pictures (crb). 79 Alamy (crb). 223 Alamy Stock Photo: Chronicle (cla). 224
Stock Photo: Granger Historical Picture Archive (b). 81 Dorling Kindersley: The Science Museum (tl). Getty
Alamy Stock Photo: The History Collection (cra). 82 Images: Science & Society Picture Library (bc). 225 The
Alamy Stock Photo: Art Collection 2 (cr). 89 Alamy New York Public Library: (tr). 227 Getty Images:
Stock Photo: The History Collection (bl). 90 Alamy Stocktrek Images (cb). SuperStock: Fine Art Images /
Stock Photo: Ian Robinson (clb). 91 SuperStock: fototeca A. Burkatovski (tr). 229 Daina Taimina: From Daina
gilardi / Marka (crb). 94 SuperStock: Melvyn Longhurst Taimina’s book Crocheting Adventures with Hyperbolic
(bl). 98 Getty Images: DEA PICTURE LIBRARY / De Planes (cb); Tom Wynne (tr). 231 Getty Images: Bettmann
Agostini (t). 99 Getty Images: DEA / M. SEEMULLER / (tr). 232 Getty Images: Boston Globe / Rubik’s Cube®
De Agostini (br). 103 Bridgeman Images: Pictures from used by permission of Rubik’s Brand Ltd www.rubiks.com
History (cra). 105 Alamy Stock Photo: Idealink (bc). 233 Alamy Stock Photo: Massimo Dallaglio (tl).
Photography (cra). 108 Alamy Stock Photo: David Lyons 235 Alamy Stock Photo: The History Collection (tr);
(bl). 110 Alamy Stock Photo: Acorn 1 (bc). 111 Alamy Jochen Tack (cla). 237 Alamy Stock Photo: Painters (tr).
Stock Photo: Flhc 80 (clb); RayArt Graphics (tr). 112 239 Alamy Stock Photo: Beth Dixson (bl). Wellcome
Getty Images: Smith Collection / Gado / Archive Photos Collection http://creativecommons.org/licenses/
(bc). 121 Alamy Stock Photo: Art Collection 3 (bl). 122 by/4.0/: (tr). 245 Wellcome Collection http://
Alamy Stock Photo: Painting (t). 123 Dreamstime. creativecommons.org/licenses/by/4.0/: (bl). 247
com: Millafedotova (crb). 126 Alamy Stock Photo: Alamy Stock Photo: sciencephotos (tl). 249 Alamy
James Davies (bc). 127 Getty Images: David Williams / Stock Photo: Chronicle (tr); Peter Horree (cla). 251 Alamy
Photographer’s Choice RF (tr). 131 Science Photo Stock Photo: The History Collection (tr). Science Photo