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The Math Book, Big Ideas Simply Explained

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3

CONTENTS
HOW TO USE THIS EBOOK
INTRODUCTION
ANCIENT AND CLASSICAL PERIODS 6000 BCE–500 CE
Numerals take their places • Positional numbers
The square as the highest power • Quadratic equations
The accurate reckoning for inquiring into all things • The Rhind papyrus
The sum is the same in every direction • Magic squares
Number is the cause of gods and daemons • Pythagoras
A real number that is not rational • Irrational numbers
The quickest runner can never overtake the slowest • Zeno’s paradoxes of motion
Their combinations give rise to endless complexities • The Platonic solids
Demonstrative knowledge must rest on necessary basic truths • Syllogistic logic
The whole is greater than the part • Euclid’s Elements
Counting without numbers • The abacus
Exploring pi is like exploring the Universe • Calculating pi
We separate the numbers as if by some sieve • Eratosthenes’ sieve
A geometrical tour de force • Conic sections
The art of measuring triangles • Trigonometry
Numbers can be less than nothing • Negative numbers
The very flower of arithmetic • Diophantine equations
An incomparable star in the firmament of wisdom • Hypatia
The closest approximation of pi for a millennium • Zu Chongzhi

THE MIDDLE AGES 500–1500
A fortune subtracted from zero is a debt • Zero
Algebra is a scientific art • Algebra
Freeing algebra from the constraints of geometry • The binomial theorem
Fourteen forms with all their branches and cases • Cubic equations
The ubiquitous music of the spheres • The Fibonacci sequence

4

The power of doubling • Wheat on a chessboard

THE RENAISSANCE 1500–1680
The geometry of art and life • The golden ratio
Like a large diamond • Mersenne primes
Sailing on a rhumb • Rhumb lines
A pair of equal-length lines • The equals sign and other symbology
Plus of minus times plus of minus makes minus • Imaginary and complex numbers
The art of tenths • Decimals
Transforming multiplication into addition • Logarithms
Nature uses as little as possible of anything • The problem of maxima
The fly on the ceiling • ; Coordinates
A device of marvelous invention • The area under a cycloid
Three dimensions made by two • Projective geometry
Symmetry is what we see at a glance • Pascal’s triangle
Chance is bridled and governed by law • Probability
The sum of the distance equals the altitude • Viviani’s triangle theorem
The swing of a pendulum • Huygens’s tautochrone curve
With calculus I can predict the future • Calculus
The perfection of the science of numbers • Binary numbers

THE ENLIGHTENMENT 1680–1800
To every action there is an equal and opposite reaction • Newton’s laws of motion
Empirical and expected results are the same • The law of large numbers
One of those strange numbers that are creatures of their own • Euler’s number
Random variation makes a pattern • Normal distribution
The seven bridges of Königsberg • Graph theory
Every even integer is the sum of two primes • The Goldbach conjecture
The most beautiful equation • Euler’s identity
No theory is perfect • Bayes’ theorem
Simply a question of algebra • The algebraic resolution of equations
Let us gather facts • Buffon’s needle experiment
Algebra often gives more than is asked of her • The fundamental theorem of algebra

5

THE 19TH CENTURY 1800–1900
Complex numbers are coordinates on a plane • The complex plane
Nature is the most fertile source of mathematical discoveries • Fourier analysis
The imp that knows the positions of every particle in the Universe • Laplace’s demon
What are the chances? • The Poisson distribution
An indispensable tool in applied mathematics • Bessel functions
It will guide the future course of science • The mechanical computer
A new kind of function • Elliptic functions
I have created another world out of nothing • Non-Euclidean geometries
Algebraic structures have symmetries • Group theory
Just like a pocket map • Quaternions
Powers of natural numbers are almost never consecutive • Catalan’s conjecture
The matrix is everywhere • Matrices
An investigation into the laws of thought • Boolean algebra
A shape with just one side • The Möbius strip
The music of the primes • The Riemann hypothesis
Some infinities are bigger than others • Transfinite numbers
A diagrammatic representation of reasonings • Venn diagrams
The tower will fall and the world will end • The Tower of Hanoi
Size and shape do not matter, only connections • Topology
Lost in that silent, measured space • The prime number theorem

MODERN MATHEMATICS 1900–PRESENT
The veil behind which the future lies hidden • 23 problems for the 20th century
Statistics is the grammar of science • The birth of modern statistics
A freer logic emancipates us • The logic of mathematics
The Universe is four-dimensional • Minkowski space
Rather a dull number • Taxicab numbers
A million monkeys banging on a million typewriters • The infinite monkey theorem
She changed the face of algebra • Emmy Noether and abstract algebra
Structures are the weapons of the mathematician • The Bourbaki group
A single machine to compute any computable sequence • The Turing machine

6

Small things are more numerous than large things • Benford’s law
A blueprint for the digital age • Information theory
We are all just six steps away from each other • Six degrees of separation
A small positive vibration can change the entire cosmos • The butterfly effect
Logically things can only partly be true • Fuzzy logic
A grand unifying theory of mathematics • The Langlands Program
Another roof, another proof • Social mathematics
Pentagons are just nice to look at • The Penrose tile
Endless variety and unlimited complication • Fractals
Four colors but no more • The four-color theorem
Securing data with a one-way calculation • Cryptography
Jewels strung on an as-yet invisible thread • Finite simple groups
A truly marvelous proof • Proving Fermat’s last theorem
No other recognition is needed • Proving the Poincaré conjecture

DIRECTORY
GLOSSARY
CONTRIBUTORS
QUOTATIONS
ACKNOWLEDGMENTS
COPYRIGHT

7

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8

FOREWORD
Summarizing all of mathematics in one book is a daunting and indeed impossible
task. Humankind has been exploring and discovering mathematics for millennia.
Practically, we have relied on math to advance our species, with early arithmetic
and geometry providing the foundations for the first cities and civilizations. And
philosophically, we have used mathematics as an exercise in pure thought to
explore patterns and logic.
As a subject, mathematics is surprisingly hard to pin down with one catch-all
definition. “Mathematics” is not simply, as many people think, “stuff to do with
numbers.” That would exclude a huge range of mathematical topics, including
much of the geometry and topology covered in this book. Of course, numbers are
still very useful tools to understand even the most esoteric areas of mathematics,
but the point is that they are not the most interesting aspect of it. Focusing just on
numbers misses the forest for the threes.
For the record, my own definition of math as “the sort of things that
mathematicians enjoy doing,” while delightfully circular, is largely unhelpful. Big
Ideas Simply Explained is actually not a bad definition. Mathematics could be
seen as the attempt to find the simplest explanations for the biggest ideas. It is the
endeavor of finding and summarizing patterns. Some of those patterns involve the
practical triangles required to build pyramids and divide land; other patterns
attempt to classify all of the 26 sporadic groups of abstract algebra. These are
very different problems in terms of both usefulness and complexity, but both
types of pattern have become the obsession of mathematicians throughout the
ages.
There is no definitive way to organize all of mathematics, but looking at it
chronologically is not a bad way to go. This book uses the historical journey of
humans discovering math as a way to classify it and wrangle it into a linear
progression, which is a valiant but difficult effort. Our current mathematical body
of knowledge has been built up by a haphazard and diverse group 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 where the
sum in each row, column, and diagonal is always the same—are one of the oldest

9

areas of recreational mathematics. Starting in the 9th century BCE in China, the
story then bounces around via Indian texts from 100 CE, Arab scholars in the
Middle Ages, Europe during the Renaissance, and finally modern Sudoku-style
puzzles. Across a mere two pages this book has to cover 3,000 years of history
ending with geomagic squares in 2001. And even in this small niche of
mathematics, there are many magic square developments that there was simply
not enough room to include. The whole book should be viewed as a curated tour
of mathematical highlights.
Studying even just a sample of mathematics is a great reminder of how much
humans have achieved. But it also highlights where mathematics could do better;
things like the glaring omission of women from the history of mathematics cannot
be ignored. A lot of talent has been squandered over the centuries, and a lot of
credit has not been appropriately given. But I hope that we are now improving the
diversity of mathematicians and encouraging all humans to discover and learn
about mathematics.
Because going forward, the body of mathematics will continue to grow. Had this
book been written a century earlier it would have been much the same up until
about page 280. And then it would have ended. No ring theory from Emmy
Noether, no computing from Alan Turing, and no six degrees of separation from
Kevin Bacon. And no doubt that will be true again 100 years from now. The
edition printed a century from now will carry on past page 325, covering patterns
totally alien to us. And because anyone can do math, there is no telling who will
discover this new math, and where or when. To make the biggest advancement in
mathematics during the 21st century, we need to include all people. I hope this
book helps inspire everyone to get involved.

Matt Parker

10

11

INTRODUCTION
The history of mathematics reaches back to prehistory, when early humans found
ways to count and quantify things. In doing so, they began to identify certain
patterns and rules in the concepts of numbers, sizes, and shapes. They discovered
the basic principles of addition and subtraction—for example, that two things
(whether pebbles, berries, or mammoths) when added to another two invariably
resulted in four things. While such ideas may seem obvious to us today, they were
profound insights for their time. They also demonstrate that the history of
mathematics is above all a story of discovery rather than invention. Although it
was human curiosity and intuition that recognized the underlying principles of
mathematics, and human ingenuity that later provided various means of recording
and notating them, those principles themselves are not a human invention. The
fact that 2 + 2 = 4 is true, independent of human existence; the rules of
mathematics, like the laws of physics, are universal, eternal, and unchanging.
When mathematicians first showed that the angles of any triangle in a flat plane
when added together come to 180°, a straight line, this was not their invention:
they had simply discovered a fact that had always been (and will always be) true.

Early applications
The process of mathematical discovery began in prehistoric times, with the
development of ways of counting things people needed to quantify. At its
simplest, this was done by cutting tally marks in a bone or stick, a rudimentary
but reliable means of recording numbers of things. In time, words and symbols
were assigned to the numbers and the first systems of numerals began to evolve, a

12

means of expressing operations such as acquisition of additional items, or
depletion of a stock, the basic operations of arithmetic.
As hunter-gatherers turned to trade and farming, and societies became more
sophisticated, arithmetical operations and a numeral system became essential
tools in all kinds of transactions. To enable trade, stocktaking, and taxes in
uncountable goods such as oil, flour, or plots of land, systems of measurement
were developed, putting a numerical value on dimensions such as weight and
length. Calculations also became more complex, developing the concepts of
multiplication and division from addition and subtraction—allowing the area of
land to be calculated, for example.
In the early civilizations, these new discoveries in mathematics, and specifically
the measurement of objects in space, became the foundation of the field of
geometry, knowledge that could be used in building and toolmaking. In using
these measurements for practical purposes, people found that certain patterns
were emerging, which could in turn prove useful. A simple but accurate
carpenter’s square can be made from a triangle with sides of three, four, and five
units. Without that accurate tool and knowledge, the roads, canals, ziggurats, and
pyramids of ancient Mesopotamia and Egypt could not have been built. As new
applications for these mathematical discoveries were found—in astronomy,
navigation, engineering, bookkeeping, taxation, and so on—further patterns and
ideas emerged. The ancient civilizations each established the foundations of
mathematics through this interdependent process of application and discovery, but
also developed a fascination with mathematics for its own sake, so-called pure
mathematics. From the middle of the first millennium BCE, the first pure
mathematicians began to appear in Greece, and slightly later in India and China,
building on the legacy of the practical pioneers of the subject—the engineers,
astronomers, and explorers of earlier civilizations.
Although these early mathematicians were not so concerned with the practical
applications of their discoveries, they did not restrict their studies to mathematics
alone. In their exploration of the properties of numbers, shapes, and processes,
they discovered universal rules and patterns that raised metaphysical questions
about the nature of the cosmos, and even suggested that these patterns had
mystical properties. Often mathematics was therefore seen as a complementary
discipline to philosophy—many of the greatest mathematicians through the ages

13

have also been philosophers, and vice versa—and the links between the two
subjects have persisted to the present day.
It is impossible to be a mathematician without being a poet of the soul.
Sofya Kovalevskaya
Russian mathematician

Arithmetic and algebra
So began the history of mathematics as we understand it today—the discoveries,
conjectures, and insights of mathematicians that form the bulk of this book. As
well as the individual thinkers and their ideas, it is a story of societies and
cultures, a continuously developing thread of thought from the ancient
civilizations of Mesopotamia and Egypt, through Greece, China, India, and the
Islamic empire to Renaissance Europe and into the modern world. As it evolved,
mathematics was also seen to comprise several distinct but interconnected fields
of study.
The first field to emerge, and in many ways the most fundamental, is the study of
numbers and quantities, which we now call arithmetic, from the Greek word
arithmos (“number”). At its most basic, it is concerned with counting and
assigning numerical values to things, but also the operations, such as addition,
subtraction, multiplication, and division, that can be applied to numbers. From the
simple concept of a system of numbers comes the study of the properties of
numbers, and even the study of the very concept itself. Certain numbers—such as
the constants π, e, or the prime and irrational numbers—hold a special fascination
and have become the subject of considerable study.

14

Another major field in mathematics is algebra, which is the study of structure,
the way that mathematics is organized, and therefore has some relevance in every
other field. What marks algebra from arithmetic is the use of symbols, such as
letters, to represent variables (unknown numbers). In its basic form, algebra is the
study of the underlying rules of how those symbols are used in mathematics—in
equations, for example. Methods of solving equations, even quite complex
quadratic equations, had been discovered as early as the ancient Babylonians, but
it was medieval mathematicians of the Islamic Golden Age who pioneered the use
of symbols to simplify the process, giving us the word “algebra,” which is derived
from the Arabic al-jabr. More recent developments in algebra have extended the
idea of abstraction into the study of algebraic structure, known as abstract algebra.
Geometry is knowledge of the eternally existent.
Pythagoras
Ancient Greek mathematician

Geometry and calculus
A third major field of mathematics, geometry, is concerned with the concept of
space, and the relationships of objects in space: the study of the shape, size, and
position of figures. It evolved from the very practical business of describing the
physical dimensions of things, in engineering and construction projects,
measuring and apportioning plots of land, and astronomical observations for
navigation and compiling calendars. A particular branch of geometry,
trigonometry (the study of the properties of triangles), proved to be especially
useful in these pursuits. Perhaps because of its very concrete nature, for many
ancient civilizations, geometry was the cornerstone of mathematics, and provided
a means of problem-solving and proof in other fields.
This was particularly true of ancient Greece, where geometry and mathematics
were virtually synonymous. The legacy of great mathematical philosophers such
as Pythagoras, Plato, and Aristotle was consolidated by Euclid, whose principles
of mathematics based on a combination of geometry and logic were accepted as
the subject’s foundation for some 2,000 years. In the 1800s, however, alternatives
to classical Euclidean geometry were proposed, opening up new areas of study,
including topology, which examines the nature and properties not only of objects
in space, but of space itself.
15

Since the Classical period, mathematics had been concerned with static
situations, or how things are at any given moment. It failed to offer a means of
measuring or calculating continuous change. Calculus, developed independently
by Gottfried Leibniz and Isaac Newton in the 1600s, provided an answer to this
problem. The two branches of calculus, integral and differential, offered a method
of analyzing such things as the slope of curves on a graph and the area beneath
them as a way of describing and calculating change.
The discovery of calculus opened up a field of analysis that later became
particularly relevant to, for example, the theories of quantum mechanics and
chaos theory in the 1900s.

Revisiting logic
The late 19th and early 20th centuries saw the emergence of another field of
mathematics—the foundations of mathematics. This revived the link between
philosophy and mathematics. Just as Euclid had done in the 3rd century BCE,
scholars including Gottlob Frege and Bertrand Russell sought to discover the
logical foundations on which mathematical principles are based. Their work
inspired a re-examination of the nature of mathematics itself, how it works, and
what its limits are. This study of basic mathematical concepts is perhaps the most
abstract field, a sort of meta-mathematics, yet an essential adjunct to every other
field of modern mathematics.
In mathematics, the art of asking questions is more valuable than solving problems.
Georg Cantor
German mathematician

16

New technology, new ideas
The various fields of mathematics—arithmetic, algebra, geometry, calculus, and
foundations—are worthy of study for their own sake, and the popular image of
academic mathematics is that of an almost incomprehensible abstraction. But
applications for mathematical discoveries have usually been found, and advances
in science and technology have driven innovations in mathematical thinking.
A prime example is the symbiotic relationship between mathematics and
computers. Originally developed as a mechanical means of doing the “donkey
work” of calculation to provide tables for mathematicians, astronomers and so on,
the actual construction of computers required new mathematical thinking. It was
mathematicians, as much as engineers, who provided the means of building
mechanical, and then electronic computing devices, which in turn could be used
as tools in the discovery of new mathematical ideas. No doubt, new applications
for mathematical theorems will be found in the future too—and with numerous
problems still unsolved, it seems that there is no end to the mathematical
discoveries to be made.
The story of mathematics is one of exploration of these different fields, and the
discovery of new ones. But it is also the story of the explorers, the
mathematicians who set out with a definite aim in mind, to find answers to
unsolved problems, or to travel into unknown territory in search of new ideas—
and those who simply stumbled upon an idea in the course of their mathematical
journey, and were inspired to see where it would lead. Sometimes the discovery
would come as a game-changing revelation, providing a way into unexplored
fields; at other times it was a case of “standing on the shoulders of giants,”

17

developing the ideas of previous thinkers, or finding practical applications for
them.
This book presents many of the “big ideas” in mathematics, from the earliest
discoveries to the present day, explaining them in layperson’s language, where
they came from, who discovered them, and what makes them significant. Some
may be familiar, others less so. With an understanding of these ideas, and an
insight into the people and societies in which they were discovered, we can gain
an appreciation of not only the ubiquity and usefulness of mathematics, but also
the elegance and beauty that mathematicians find in the subject.
Mathematics, rightly viewed, possesses not only truth, but supreme beauty.
Bertrand Russell
British philosopher and mathematician

18

19

INTRODUCTION
As early as 40,000 years ago, humans were making tally marks on wood and bone
as a means of counting. They undoubtedly had a rudimentary sense of number
and arithmetic, but the history of mathematics only properly began with the
development of numerical systems in early civilizations. The first of these
emerged in the sixth millennium BCE, in Mesopotamia, western Asia, home to the
world’s earliest agriculture and cities. Here, the Sumerians elaborated on the
concept of tally marks, using different symbols to denote different quantities,
which the Babylonians then developed into a sophisticated numerical system of
cuneiform (wedge-shaped) characters. From about 4000 BCE, the Babylonians
used elementary geometry and algebra to solve practical problems—such as
building, engineering, and calculating land divisions—alongside the arithmetical
skills they used to conduct commerce and levy taxes.
A similar story emerges in the slightly later civilization of the ancient Egyptians.
Their trade and taxation required a sophisticated numerical system, and their
building and engineering works relied on both a means of measurement and some
knowledge of geometry and algebra. The Egyptians were also able to use their
mathematical skills in conjunction with observations of the heavens to calculate
and predict astronomical and seasonal cycles and construct calendars for the
religious and agricultural year. They established the study of the principles of
arithmetic and geometry as early as 2000 BCE.

Greek rigor
The 6th century BCE onward saw a rapid rise in the influence of ancient Greece
across the eastern Mediterranean. Greek scholars quickly assimilated the
mathematical ideas of the Babylonians and Egyptians. The Greeks used a
numerical system of base-10 (with ten symbols) derived from the Egyptians.
20

Geometry in particular chimed with Greek culture, which idolized beauty of form
and symmetry. Mathematics became a cornerstone of Classical Greek thinking,
reflected in its art, architecture, and even philosophy. The almost mystical
qualities of geometry and numbers inspired Pythagoras and his followers to
establish a cultlike community, dedicated to studying the mathematical principles
they believed were the foundations of the Universe and everything in it.
Centuries before Pythagoras, the Egyptians had used a triangle with sides of 3, 4,
and 5 units as a building tool to ensure corners were square. They had come
across this idea by observation, and then applied it as a rule of thumb, whereas the
Pythagoreans set about rigorously showing the principle, offering a proof that it is
true for all right-angled triangles. It is this notion of proof and rigor that is the
Greeks’ greatest contribution to mathematics.
Plato’s Academy in Athens was dedicated to the study of philosophy and
mathematics, and Plato himself described the five Platonic solids (the tetrahedron,
cube, octahedron, dodecahedron, and icosahedron). Other philosophers, notably
Zeno of Elea, applied logic to the foundations of mathematics, exposing the
problems of infinity and change. They even explored the strange phenomenon of
irrational numbers. Plato’s pupil Aristotle, with his methodical analysis of logical
forms, identified the difference between inductive reasoning (such as inferring a
rule of thumb from observations) and deductive reasoning (using logical steps to
reach a certain conclusion from established premises, or axioms).
From this basis, Euclid laid out the principles of mathematical proof from
axiomatic truths in his Elements, a treatise that was the foundation of mathematics
for the next two millennia. With similar rigor, Diophantus pioneered the use of
symbols to represent unknown numbers in his equations; this was the first step
toward the symbolic notation of algebra.

A new dawn in the East
Greek dominance was eventually eclipsed by the rise of the Roman Empire. The
Romans regarded mathematics as a practical tool rather than worthy of study. At
the same time, the ancient civilizations of India and China independently
developed their own numerical systems. Chinese mathematics in particular
flourished between the 2nd and 5th centuries CE, thanks largely to the work of Liu
Hui in revising and expanding the classic texts of Chinese mathematics.

21

IN CONTEXT
KEY CIVILIZATION
Babylonians
FIELD
Arithmetic
BEFORE
40,000 years ago Stone Age people in Europe and Africa count using tally
marks on wood or bone.
6000–5000 BCE Sumerians develop early calculation systems to measure land
and to study the night sky.
4000–3000 BCE Babylonians use a small clay cone for 1 and a large cone for 60,
along with a clay ball for 10, as their base-60 system evolves.
AFTER
2nd century CE The Chinese use an abacus in their base-10 positional number
system.
7th century In India, Brahmagupta establishes zero as a number in its own right
and not just as a placeholder.
It is given to us to calculate, to weigh, to measure, to observe; this is natural philosophy.
Voltaire
French philosopher

22

The first people known to have used an advanced numeration system were the
Sumerians of Mesopotamia, an ancient civilization living between the Tigris and
Euphrates rivers in what is present-day Iraq. Sumerian clay tablets from as early
as the 6th millennium BCE include symbols denoting different quantities. The
Sumerians, followed by the Babylonians, needed efficient mathematical tools in
order to administer their empires.
What distinguished the Babylonians from neighbors such as Egypt was their use
of a positional (place value) number system. In such systems, the value of a
number is indicated both by its symbol and its position. Today, for instance, in the
decimal system, the position of a digit in a number indicates whether its value is
in ones (less than 10), tens, hundreds, or more. Such systems make calculation
more efficient because a small set of symbols can represent a huge range of
values. By contrast, the ancient Egyptians used separate symbols for ones, tens,
hundreds, thousands, and above, and had no place value system. Representing
larger numbers could require 50 or more hieroglyphs.

Using different bases
The Hindu–Arabic numeration that is employed today is a base-10 (decimal)
system. It requires only 10 symbols—nine digits (1, 2, 3, 4, 5, 6, 7, 8, 9) and a
zero as a placeholder. As in the Babylonian system, the position of a digit
indicates its value, and the smallest value digit is always to the right. In a base-10
23

system, a two-digit number, such as 22, indicates (2 × 101) + 2; the value of the 2
on the left is ten times that of the 2 on the right. Placing digits after the number 22
will create hundreds, thousands, and larger powers of 10. A symbol after a whole
number (the standard notation now is a decimal point) can also separate it from its
fractional parts, each representing a tenth of the place value of the preceding
figure. The Babylonians worked with a more complex sexagesimal (base-60)
number system that was probably inherited from the earlier Sumerians and is still
used across the world today for measuring time, degrees in a circle (360° = 6 ×
60), and geographic coordinates. Why they used 60 as a number base is still not
known for sure. It may have been chosen because it can be divided by many other
numbers—1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The Babylonians also based their
calendar year on the solar year (365.24 days); the number of days in a year was
360 (6 × 60) with additional days for festivals.
In the Babylonian sexagesimal system, a single symbol was used alone and
repeated up to nine times to represent symbols for 1 to 9. For 10, a different
symbol was used, placed to the left of the one symbol, and repeated two to five
times in numbers up to 59. At 60 (60 × 1), the original symbol for one was reused
but placed further to the left than the symbol for 1. Because it was a base-60
system, two such symbols signified 61, while three such symbols indicated 3,661,
that is, 60 × 60 (602) + 60 + 1.
The base-60 system had obvious drawbacks. It necessarily requires many more
symbols than a base-10 system. For centuries, the sexagesimal system also had no
place value holders, and nothing to separate whole numbers from fractional parts.
By around 300 BCE, however, the Babylonians used two wedges to indicate no
value, much as we use a placeholder zero today; this was possibly the earliest use
of zero.

24

The Babylonian sun-god Shamash awards a rod and a coiled rope, ancient measuring
devices, to newly trained surveyors, on a clay tablet dating from around 1000 BCE.

Other counting systems
In Mesoamerica, on the other side of the world, the Mayan civilization developed
its own advanced numeration system in the 1st millennium BCE—apparently in
complete isolation. Theirs was a base-20 (vigesimal) number system, which
probably evolved from a simple counting method using fingers and toes. In fact,
base-20 number systems were used across the world, in Europe, Africa, and Asia.
Language often contains remnants of this system. For example, in French, 80 is
expressed as quatre-vingt (4 × 20); Welsh and Irish also express some numbers as
multiples of 20, while in English a score is 20. In the Bible, for instance, Psalm 90
talks of a human lifespan being “threescore years and ten” or as great as
“fourscore years.”
From around 500 BCE until the 16th century when Hindu–Arabic numbers were
officially adopted in China, the Chinese used rod numerals to represent numbers.
This was the first decimal place value system. By alternating quantities of vertical
rods with horizontal rods, this system could indicate ones, tens, hundreds,
thousands, and more powers of 10, much as the decimal system does today. For
25

example, 45 was written with four horizontal bars representing 4 × 101 (40) and
five vertical bars for 5 × 1 (5). However, four vertical rods followed by five
vertical rods indicated 405 (4 × 100, or 102) + 5 × 1—the absence of horizontal
rods meant there were no tens in the number. Calculations were carried out by
manipulating the rods on a counting board. Positive and negative numbers were
represented by red and black rods respectively or different cross sections
(triangular and rectangular). Rod numerals are still used occasionally in China,
just as Roman numerals are sometimes used in Western society.
The Chinese place value system is reflected in the Chinese abacus (suanpan).
Dating back to at least 200 BCE, it is one of the oldest bead-counting devices,
although the Romans used something similar. The Chinese version, which is still
used today, has a central bar and a varying number of vertical wires to separate
ones from tens, hundreds, or more. In each column, there are two beads above the
bar worth five each and five beads below the bar worth one each.
The Japanese adopted the Chinese abacus in the 14th century and developed their
own abacus, the soroban, which has one bead worth five above the central bar and
four beads each worth one below the bar in each column. Japan still uses the
soroban today: there are even contests in which young people demonstrate their
ability to perform soroban calculations mentally, a skill known as anzan.

Cuneiform

Cuneiform, a word
derived from the Latin
cuneus (“wedge”) to
describe the shape of the
symbols, was inscribed
into wet clay, stone, or
metal.

In the late 1800s, academics deciphered the
“cuneiform” (wedge-shaped) markings on clay
tablets recovered from Babylonian sites in and
around Iraq. Such marks, denoting letters and words
as well as an advanced number system, were etched
in wet clay with either end of a stylus. Like the
Egyptians, the Babylonians needed scribes to
administer their complex society, and many of the
tablets bearing mathematical records are thought to
be from training schools for scribes.
A great deal has now been discovered about
Babylonian mathematics, which extended to
multiplication, division, geometry, fractions, square
roots, cube roots, equations, and other forms,
26

because—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.

The Babylonian base-60 number system was built from two symbols—the single unit
symbol, used alone and combined for numbers 1 to 9, and the 10 symbol, repeated for 20, 30,
40, and 50.
The Babylonian and Assyrian civilizations have perished…yet Babylonian mathematics is still
interesting, and the Babylonian scale of 60 is still used in astronomy.
G. H. Hardy
British mathematician

Modern numeration
The Hindu–Arabic decimal system used throughout the world today has its
origins in India. In the 1st to 4th centuries CE, the use of nine symbols along with
zero was developed to allow any number to be written efficiently, through the use
of place value. The system was adopted and refined by Arab mathematicians in
the 9th century. They introduced the decimal point, so that the system could also
express fractions of whole numbers.

27

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 whether to use the new system rather than Roman numerals and
traditional counting methods lasted for several hundred years, before its adoption
paved the way for modern mathematical advances.
With the advent of electronic computers, other number bases became important
—particularly binary, a number system with base 2. Unlike the base-10 system
with its 10 symbols, binary has just two: 1 and 0. It is a positional system, but
instead of multiplying by 10, each column is multiplied by 2, also expressed as 21,
22, 23 and upward. In binary, the number 111 means 1 × 22 + 1 × 21 + 1 × 20, that
is 4 + 2 + 1, or 7 in our decimal number system.
In binary, as in all modern number systems whatever their base, the principles of
place value are always the same. Place value—the Babylonian legacy—remains a
powerful, easily understood, and efficient way to represent large numbers.
The fact that we work in 10s as opposed to any other number is purely a consequence of our
anatomy. We use our ten fingers to count.
Marcus du Sautoy
British mathematician

28

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.

Mayan numeral system

The Dresden Codex, the
oldest surviving Mayan
book, dating from the 13th
or 14th century, illustrates

The Mayans, who lived in Central America from
around 2000 BCE, used a base-20 (vigesimal) number
system from around 1000 BCE to perform
astronomical and calendar calculations. Like the
Babylonians, they used a calendar of 360 days plus
festivals, to make 365.24 days based on the solar
year; their calendars helped them work out the
growing cycles of crops.
The Mayan system employed symbols: a dot
representing one and a bar representing five. By
using combinations of dots over bars they could
29

Mayan number symbols
and glyphs.

generate numerals up to 19. Numbers larger than 19
were written vertically, with the lowest numbers at
the bottom, and there is evidence of Mayan
calculations up to hundreds of millions. An inscription from 36 BCE shows that
they used a shell-shaped symbol to denote zero, which was widely used by the
4th century.
The Mayans’ number system was in use in Central America until the Spanish
conquests in the 16th century. Its influence, however, never spread further.
See also: The Rhind papyrus • The abacus • Negative numbers • Zero • The
Fibonacci sequence • Decimals

30

IN CONTEXT
KEY CIVILIZATIONS
Egyptians (c. 2000 BCE), Babylonians (c. 1600 BCE)
FIELD
Algebra
BEFORE
c. 2000 BCE The Berlin papyrus records a quadratic equation solved in ancient
Egypt.
AFTER
7th century CE The Indian mathematician Brahmagupta solves quadratic
equations using only positive integers.
10th century CE Egyptian scholar Abu Kamil Shuja ibn 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.
Quadratic 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.

31

Ancient roots
The history of quadratic equations extends across the world. It is likely that these
equations first arose from the need to subdivide land for inheritance purposes, or
to solve problems involving addition and multiplication.
One of the oldest surviving examples of a quadratic equation comes from the
ancient Egyptian text known as the Berlin papyrus (c. 2000 BCE). The problem
contains the following information: the area of a square of 100 cubits is equal to
that of two smaller squares. The side of one of the smaller squares is equal to one
half plus a quarter of the side of the other. In modern notation, this translates into
two simultaneous equations: x2 + y2 = 100 and x = (1⁄2 + 1⁄4)y = 3⁄4 y. These can be
simplified to the quadratic equation (3⁄4 y)2 + y2 = 100 to find the length of a side
on each square.
The Egyptians used a method called “false position” to determine the solution. In
this method, the mathematician selects a convenient number that is usually easy to
calculate, then works out what the solution to the equation would be using that
32

number. The result shows how to adjust the number to give the correct solution
the equation. For example, in the Berlin papyrus problem, the simplest length to
use for the larger of the two small squares is 4, because the problem deals with
quarters. For the side of the smallest square, 3 is used because this length is 3⁄4 of
the side of the other small square. Two squares created using these false position
numbers would have areas of 16 and 9 respectively, which when added together
give a total area of 25. This is only 1⁄4 of 100, so the areas must be quadrupled to
match the Berlin papyrus equation. The lengths therefore must be doubled from
the false positions of 4 and 3 to reach the solutions: 8 and 6.
Other early records of quadratic equations are found in Babylonian clay tablets,
where the diagonal of a square is given to five decimal places. The Babylonian
tablet YBC 7289 (c. 1800–1600 BCE) shows a method of working out the
quadratic equation x2 = 2 by drawing rectangles and trimming them down into
squares. In the 7th century CE, Indian mathematician Brahmagupta wrote a
formula for solving quadratic equations that could be applied to equations in the
form ax2 + bx = c. Mathematicians at the time did not use letters or symbols, so
he wrote his solution in words, but it was similar to the modern formula shown
above.
In the 8th century, Persian mathematician al-Khwarizmi employed a geometric
solution for quadratic equations known as completing the square. Until the 10th
century, geometric methods were were often used, as quadratic equations were
used to solve real-world problems involving land rather than abstract algebraic
challenges.

33

The Berlin papyrus was copied and published by German Egyptologist Hans SchackSchackenburg in 1900. It contains two mathematical problems, one of which is a quadratic
equation.

Negative solutions
Indian, Persian, and Arab scholars thus far had used only positive numbers. When
solving the equation x2 + 10x = 39, they gave the solution as 3. However, this is
one of two correct solutions to the problem; -13 is the other. If x is -13, x2 = 169
and 10x = -130. Adding a negative number gives the same result as subtracting its
equivalent positive number, so 169 + -130 = 169 - 130 = 39.
In the 10th century, Egyptian scholar Abu Kamil Shuja ibn Aslam made use of
negative numbers and algebraic irrational numbers (such as the square root of 2)

34

as both solutions and coefficients (numbers multiplying an unknown quantity).
By the 1500s, most mathematicians accepted negative solutions and were
comfortable with surds (irrational roots – those that cannot be expressed exactly
as a decimal). They had also started using numbers and symbols, rather than
writing equations in words. Mathematicians now utilized the plus or minus
symbol, ±, in solving quadratic equations. With the equation x2 = 2, the solution
is not just x =

but x = ±

. The plus or minus symbol is included because

two negative numbers multiplied together make a positive number. While
= 2, it is also true that -

×-

×

= 2.

In 1545, Italian scholar Gerolamo Cardano published his Ars Magna (The Great
Art, or the Rules of Algebra) in which he explored the problem: “What pair of
numbers have a sum of ten and product of 40?” He found that the problem led to
. No
a quadratic equation which, when he completed the square, gave
numbers available to mathematicians at the time gave a negative number when
multiplied by themselves, but Cardano suggested suspending belief and working
with the square root of negative 15 to find the equation’s two solutions. Numbers
would later be known as “imaginary” numbers.
such as

The quadratic formula is a way to solve quadratic equations. By modern convention,
quadratic equations include a number, a, multiplied by x2; a number, b, multiplied by x; and a
number, c, on its own. The illustration above 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 out
on a graph; the x solutions are the points where the curve crosses the x axis.
Politics is for the present, but an equation is for eternity.
Albert Einstein

35

Structure of equations
Modern quadratic equations usually look like ax2 + bx + c = 0. The letters a, b,
and c represent known numbers, while x represents the unknown number.
Equations contain variables (symbols for numbers that are unknown),
coefficients, constants (those that do not multiply variables), and operators
(symbols such as the plus and equals sign). Terms are the parts separated by
operators; they can be a number or variable, or a combination of both. The
modern quadratic equation has four terms: ax2, bx, c, and 0.

A graph of the quadratic function y = ax2 + bx + c creates a U-shaped curve called a
parabola. This graph plots the points (in black) of the quadratic function where a = 1, b = 3,

36

and c = 2. This expresses the quadratic equation x2 + 3x + 2 = 0. The solutions for x are where
y = 0 and the curve crosses the x axis. These are -2 and -1.

Parabolas
A function is a group of terms that defines a relationship between variables (often
x and y). The quadratic function is generally written as y = ax2 + bx + c, which,
on a graph, produces a curve called a parabola. When real (not imaginary)
solutions to ax2 + bx + c = 0 exist, they will be the roots—the points where the
parabola crosses the x axis. Not all parabolas cut the x axis in two places. If the
parabola touches the x axis only once, this means that there are coincident roots
(the solutions are equal to each other). The simplest equation of this form is y =
x2. If the parabola does not touch or cross the x axis, there are no real roots.
Parabolas prove useful in the real world because of their reflective. properties.
Satellite dishes are parabolic for this reason. Signals received by the dish will
reflect off the parabola and be directed to one single point—the receiver.

Parabolic objects have special reflective properties. With a parabolic mirror, any ray of light
parallel to its line of symmetry will reflect off the surface to the same fixed point (A).

Practical applications
37

Quadratic equations are
used by military
specialists to model the
trajectory of projectiles
fired by artillery—such as
this MIM-104 Patriot
surface-to-air missile,
commonly used by the US
Army.

Although they were initially used for working out
geometric problems, today quadratic equations are
important in many aspects of mathematics, science,
and technology. Projectile flight, for example, can
be modeled with quadratic equations. An object
thrown up into the air will fall back down again as a
result of gravity. The quadratic function can predict
projectile motion—the height of the object over
time. Quadratic equations are used to model the
relationship between time, speed, and distance, and
in calculations with parabolic objects such as lenses.
They can also be used to forecast profits and loss in
the world of business. Profit is based on total
revenue minus production cost; companies create a
quadratic equation known as the profit function with
these variables to work out the optimal sale prices to

maximize profits.
See also: Irrational numbers • Negative numbers • Diophantine equations • Zero •
Algebra • The binomial theorem • Cubic equations • Imaginary and complex
numbers

38

IN CONTEXT
KEY CIVILIZATION
Ancient Egyptians (c. 1650 BCE)
FIELD
Arithmetic
BEFORE
c. 2480 BCE Stone carvings record flood levels on the River Nile, measured in
cubits—about 201⁄2 in (52 cm)—and palms—about 3 in (7.5 cm).
c. 1800 BCE The Moscow papyrus provides solutions to 25 mathematical
problems, including the calculation of the surface area of a hemisphere and the
volume of a pyramid.
AFTER
c. 1300 BCE The Berlin papyrus is produced. It shows that the ancient Egyptians
used quadratic equations.
6th century BCE The Greek scientist Thales travels to Egypt and studies its
mathematical theories.
The Rhind papyrus in the British Museum in London provides an intriguing
account of mathematics in ancient Egypt. Named after Scottish antiquarian
Alexander Henry Rhind, who purchased the papyrus in Egypt in 1858, it was
copied from earlier documents by a scribe, Ahmose, more than 3,500 years ago. It
measures 121⁄2 in (32 cm) by 781⁄2 in (200 cm) and includes 84 problems
concerned with arithmetic, algebra, geometry, and measurement. The problems,

39

recorded in this and other ancient Egyptian artifacts such as the earlier Moscow
papyrus, illustrated techniques for working out areas, proportions, and volumes.

The Eye of Horus, an Egyptian god, was a symbol of power and protection. Parts of it were
also used to denote fractions whose denominators were powers of 2. The eyeball, for
example, represents 1⁄4, while the eyebrow is 1⁄8.

Representing concepts
The Egyptian number system was the first decimal system. It used strokes for
single digits and a different symbol for each power of 10. The symbols were then
repeated to create other numbers. A fraction was shown as a number with a dot
above it. The Egyptian concept of a fraction was closest to a unit fraction—that is,
1⁄ , where n is a whole number. When a fraction was doubled, it had to be
n
rewritten as one unit fraction added to another unit fraction; for example, 2⁄3 in
modern notation would be 1⁄2 + 1⁄6 in Egyptian notation (not 1⁄3 + 1⁄3 because the
Egyptians did not allow repeats of the same fraction).
The 84 problems in the Rhind papyrus illustrate the mathematical methods in
common use in ancient Egypt. Problem 24, for example, asks what quantity, if
added to its seventh part, becomes 19. This translates as x + x⁄7 = 19. The
approach applied to problem 24 is known as “false position.” This technique—
used well into the Middle Ages—is based on trial and improvement, choosing the
simplest, or “false,” value for a variable and adjusting the value using a scaling
factor (the required quantity divided by the result).

40

In the workings for problem 24, one-seventh is easiest to find for the number 7,
so 7 is used first as a “false” value for the variable. The result of the calculation—
7 plus 7⁄7 (or 1)—is 8, not 19, so a scaling factor is needed. To find how far the
guess of 7 is from the required quantity, 19 is divided by 8 (the “false” answer).
This produces a result of 2 + 1⁄4 + 1⁄8 (not 23⁄8, as Egyptian multiplication was
based on doubling and halving fractions), which is the scaling factor that should
be applied. So 7 (the original “false” value) is multiplied by 2 + 1⁄4 + 1⁄8 (the
scaling factor) to give the quantity 16 + 1⁄2 + 1⁄8 (or 165⁄8).
Many problems in the papyrus deal with working out shares of commodities or
land. Problem 41 asks for the volume of a cylindrical store with a diameter of 9
cubits and a height of 10 cubits. The method finds the area of a square whose side
length is 8⁄9 of the diameter, then multiplies this by the height. The figure of 8⁄9 is
used as an approximation for the proportion of the area of a square that would be
taken up by a circle if it were drawn within the square. This method is used in
problem 50 to find the area of a circle: subtract 1⁄9 from the diameter of the circle,
and find the area of the square with the resulting side length.

Ancient Egyptians used vertical lines to denote the numbers 1 to 9. Powers of 10,
particularly those inscribed on stone, were depicted as hieroglyphs—picture symbols.

Level of accuracy
Since the Ancient Greeks, the area of a circle has been found by multiplying the
square of its radius (r2) with the number pi (π), written as πr2. The ancient
Egyptians had no concept of pi, but the calculations in the Rhind papyrus show
that they were close to its value. Their circle area calculation—with the circle
41

diameter as twice the radius (2r)—can be expressed as (8⁄9 × 2r)2, which,
simplified, is 256⁄81 r2, giving an equivalent for pi of 256⁄81. As a decimal, this is
about 0.6 percent greater than the true value of pi.

Instruction books

The Rhind papyrus
scribe used the hieratic
system of writing
numerals. This cursive
style was more compact
and practical than drawing
complex hieroglyphs.

The Rhind and Moscow papyri are the most
complete mathematical documents to survive from
the height of the ancient Egyptian civilization. They
were painstakingly copied by scribes well versed in
arithmetic, geometry, and 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 were not seen as works
of scholarship. Instead, they were instruction
manuals for use in trade, accounting, construction,
and other activities that involved measurement and
calculation.

Egyptian engineers, for example, used mathematics
in the building of pyramids. The Rhind papyrus includes a calculation for the
slope of a pyramid using the seked— a measure for the horizontal distance
traveled by a slope for each drop of 1 cubit. The steeper the side of a pyramid,
the fewer the sekeds.
See also: Positional numbers • Pythagoras • Calculating pi • Algebra • Decimals

42

IN CONTEXT
KEY CIVILIZATION
Ancient Chinese
FIELD
Number theory
BEFORE
9th century BCE The Chinese I Ching (Book of Changes) lays out trigrams and
hexagrams of numbers for use in divination.
AFTER
1782 Leonhard Euler writes about Latin squares in his Recherches sur une
nouvelle espèce de carrés magiques (Investigations on a new type of magic
square).
1979 The first Sudoku-style puzzle is published by Dell Magazines in New
York.
2001 British electronics engineer Lee Sallows invents magic squares called
“geomagic squares,” which contain geometric shapes rather than numbers.
There are thousands of ways in which to arrange the numbers 1 to 9 in a three-bythree grid. Only eight of these produce a magic square, where the sum of the
numbers in each row, column, and diagonal—the magic total—is the same. The
sum of the numbers 1 to 9 is 45, as is the sum of all three rows or columns. The
magic total, therefore, is 1⁄3 of 45, or 15. In fact, there is really just one
combination of numbers in a magic square. The other seven are rotations of this
combination.
43

Ancient origins
Magic squares are probably the earliest example of “recreational mathematics.”
Their exact origin is unknown, but the first known reference, in the Chinese
legend of Lo Shu (Scroll of the river Lo), dates from 650 BCE. In the legend, a
turtle appears to the great King Yu as he faces a devastating flood. The markings
on the turtle’s back form a magic square, with numbers from 1 to 9 represented by
circular dots. Because of this legend, the arrangement of odd and even numbers
(even numbers are always in the corners of the square) were believed to have
magical properties and was used as a good luck talisman through the ages.
As ideas from China spread along trade routes such as the Silk Road, other
cultures became interested in magic squares. Magic squares are discussed in
Indian texts dating from 100 CE, and Brihat-Samhita (c. 550 CE), a book of
divination, includes the first recorded magic square in India, used to measure out
quantities of perfume. Arab scholars, who created a vital link between the
learning of ancient civilizations and the European Renaissance, introduced magic
squares to Europe in the 14th century.

44

An order-four magic square appears beneath the bell in Melencolia I by the German artist
Albrecht Dürer and wittily includes the engraving’s date of 1514.

Different-sized squares
The number of rows and columns in a magic square is called its order. For
example, a three-by-three magic square is said to have an order of three. An
order-two magic square does not exist because it would only work if all the
numbers were identical. As the orders increase, so do the quantities of magic
squares. Order four produces 880 magic squares—with a magic total of 34. There
are hundreds of millions of order-five magic squares, while the quantity of ordersix magic squares has not yet been calculated.
Magic squares have been an enduring source of fascination for mathematicians.
The 15th-century Italian mathematician Luca Pacioli, author of De viribus

45

quantitatis (On the Power of Numbers), collected magic squares. In 18th-century
Switzerland, Leonhard Euler also became interested in them, and devised a form
that he named Latin squares. The rows and columns in a Latin square contain
figures or symbols that appear only once in each row and column.
One derivation of the Latin square—Sudoku—has become a popular puzzle.
Devised in the US in the 1970s (where it was called Number Place), Sudoku took
off in Japan in the 1980s, acquiring its now-familiar name, which means “single
digits.” A Sudoku puzzle is a nine-by-nine Latin square with the added restriction
that subdivisions of the square must also contain all nine numbers.
The most magically magical of any magic square ever made by a magician.
Benjamin Franklin
Talking about a magic square that he discovered

46

Once you have one magic square, you can add the same quantity to every number in the
square and still end up with a magic square. Similarly, if you multiply all the numbers by the
same quantity, you still have a magic square.

See also: Irrational numbers • Eratosthenes’ sieve • Negative numbers • The
Fibonacci sequence • The golden ratio • Mersenne primes • Pascal’s triangle

47

IN CONTEXT
KEY FIGURE
Pythagoras (c. 570 BCE–495 BCE)
FIELD
Applied geometry
BEFORE
c. 1800 BCE The columns of cuneiform numbers on the Plimpton 322 clay tablet
from Babylon include some numbers related to Pythagorean triples.
6th century BCE Greek philosopher Thales of Miletus proposes a nonmythological explanation of the Universe— pioneering the idea that nature can
be interpreted by reason.
AFTER
c. 380 BCE In the tenth book of his Republic, Plato espouses Pythagoras’s theory
of the transmigration of souls.
c. 300 BCE Euclid produces a formula to find sets of primitive Pythagorean
triples.
The 6th-century BCE Greek philosopher Pythagoras is also antiquity’s most
famous mathematician. Whether or not he was responsible for all the many
achievements attributed to him in math, science, astronomy, music, and medicine,
there is no doubt that he founded an exclusive community that lived for the
pursuit of mathematics and philosophy, and regarded numbers as the sacred
building blocks of the Universe.

48

Thales of Miletus, one of the Seven Sages of ancient Greece, possibly inspired the younger
Pythagoras with his geometrical and scientific ideas. They may have met in Egypt.

Angles and symmetry
The Pythagoreans were masters of geometry and knew that the sum of the three
angles of a triangle (180°) is equal to the sum of two right angles (90° + 90°), a
fact which two centuries later was described by Euclid as the triangle postulate.
Pythagoras’s followers were also aware of some of the regular polyhedra; these
are the perfectly symmetrical three-dimensional shapes (such as the cube) that
were later known as the Platonic solids.
Pythagoras himself is primarily associated with the formula that describes the
relationship between the sides of a right-angled triangle. Widely known as
Pythagoras’s theorem, it states that a2 + b2 = c2, where c is the longest side of the
49

triangle (the hypotenuse), and a and b represent the other two, shorter sides that
are adjacent to the right angle. For example, a right-angled triangle with two
shorter sides of lengths 3in and 4in will have a hypotenuse of length 5in. The
length of this hypotenuse is found because 32 + 42 = 52 (9 + 16 = 25). Such sets of
whole-number solutions to the equation a2 + b2 = c2 are known as Pythagorean
triples. Multiplying the triple 3, 4, and 5 by 2 produces another Pythagorean
triple: 6, 8, and 10 (36 + 64 = 100). The set 3, 4, 5 is called a “primitive”
Pythagorean triple because its components share no common divisor larger than
1. The set 6, 8, 10 is not primitive as its components share the common divisor 2.
There is good evidence that the Babylonians and the Chinese were well aware of
the mathematical relationship between sides of a right-angled triangle centuries
before Pythagoras’s birth. However, Pythagoras is believed to have been the first
to prove the truth of the formula that states this relationship, and its validity for all
right-angled triangles, which is why the theorem takes his name.

Pythagorean triples
The sets of three integers that solve the equation a2
+ b2 = c2 are known as Pythagorean triples, although
their existence was known long before Pythagoras.
Around 1800 BCE, the Babylonians recorded sets of
Pythagorean numbers on the Plimpton 322 clay
tablet; these show that triples become more spread
out as the number line progresses. The Pythagoreans
developed methods for finding sets of triples, and
The smallest, or most
also proved that there are an infinite number of such
primitive, of the
sets. After many of Pythagoras’s schools were
Pythagorean triples is a
triangle with side lengths
destroyed in a 6th-century BCE political purge,
3, 4, and 5. As this graphic
Pythagoreans emigrated to other parts of southern
shows, 9 plus 16 equals
Italy, spreading their knowledge of triples across the
25.
ancient world. Two centuries later, Euclid developed
a formula to generate triples: a = m2 - n2, b = 2mn, c = m2 + n2. With certain
exceptions, m and n can be any two integers, such as 7 and 4, which produce the
triple 33, 56, 65 (332 + 562 = 652). The formula dramatically sped up the
process of finding new Pythagorean triples.

50

The graphic above demonstrates why the Pythagorean equation (a²+ b²= c²) works. Within a
large square there are four right-angled triangles of equal size (sides labeled 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.

Journeys of discovery
Pythagoras was well-traveled, and the ideas he absorbed from other countries
undoubtedly fueled his mathematical inspiration. Hailing from Samos, which was
not far from Miletus in western Anatolia (present-day Turkey), he may have
studied at the school of Thales of Miletus under the philosopher Anaximander. He
embarked on his travels at the age of 20, and spent many years away. He is
thought to have visited Phoenicia, Persia, Babylon, and Egypt, and may also have
reached India. The Egyptians knew that a triangle with sides of 3, 4, and 5 (the
first Pythagorean triple) would produce a right angle, so their surveyors used
ropes of these lengths to construct perfect right angles for their building projects.
Observing this method firsthand may have encouraged Pythagoras to study and
prove the underlying mathematical theorem. In Egypt, Pythagoras may also have
met Thales of Miletus, a keen geometrician, who calculated the heights of
pyramids and applied deductive reasoning to geometry.
Reason is immortal, all else is mortal.
Pythagoras

51

A Pythagorean community
After 20 years of traveling, Pythagoras eventually settled in Croton (now
Crotone), southern Italy, a city with a large Greek population. There, he
established the Pythagorean brotherhood— a community to whom he could teach
both his mathematical and philosophical beliefs. Women were welcome in the
brotherhood, and formed a significant part of its 600 members. When they joined,
members were obliged to give all their possessions and wealth to the brotherhood,
and also swore to keep its mathematical discoveries secret. Under Pythagoras’s
leadership, the community gained considerable political influence.
As well as his theorem, Pythagoras and his close-knit community made
numerous other advances in mathematics, but carefully guarded that knowledge.
Among their discoveries were polygonal numbers: these, when represented by
dots, can form the shapes of regular polygons. For 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 of the triangle (4 + 3 + 2 + 1 = 10).
Two millennia after Pythagoras, in 1638, Pierre de Fermat enlarged on this idea
when he asserted that any number could be written as the sum of up to k k-gonal
numbers; 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 be written as the sum of three triangular numbers: 1 + 3 + 15 =
19. Fermat could not prove this conjecture; it was only in 1813 that French
mathematician Augustin-Louis Cauchy completed the proof.
Strength of mind rests in sobriety; for this keeps your reason unclouded by passion.
Pythagoras

52

Fascinated by numbers
Another type of number that excited Pythagoras was the perfect number. It was so
called because it is the exact sum of all the divisors less than itself. The first
perfect number is 6, as its divisors 1, 2, and 3 add up to 6. The second is 28 (1 + 2
+ 4 + 7 + 14 = 28), the third 496, and the fourth 8,128. There was no practical
value in identifying such numbers, but their 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 π. Such numbers had no place among the wellordered 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

.

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 shape was
formed from 12 regular pentagons, and known as the dodecahedron—one of the
five Platonic solids. Pythagoreans revered the pentagon, and their symbol was the
pentagram, a five-pointed star with a pentagon at its center. Breaking the
brotherhood’s rule of secrecy by revealing their knowledge of the dodecahedron
would therefore have been an especially heinous crime, punishable by death.
The finest type of man gives himself up to discovering the meaning and purpose of life itself…
this is the man I call a philosopher.
Pythagoras

53

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.
I have often admired the mystical way of Pythagoras, and the secret magick of numbers.
Sir Thomas Browne
English polymath

An integrated philosophy
In ancient Greece, mathematics and philosophy were considered complementary
subjects and were studied together. Pythagoras is credited with coining the term
“philosopher,” from the Greek philos (“love”) and sophos (“wisdom”). For
Pythagoras and his successors, the duty of a philosopher was the pursuit of
wisdom.

54

Pythagoras’s own brand of philosophy integrated spiritual ideas with
mathematics, science, and reasoning. Among his beliefs was the idea of
metempsychosis, which he may have encountered on his travels in Egypt or
elsewhere in the Middle East. This held that souls are immortal and at death
transmigrate to occupy a new body. In Athens two centuries later, Plato was
entranced by the idea and included it in many of his dialogues. Later, Christianity,
too, embraced the idea of a division between body and soul; and Pythagoras’s
ideas would become a core part of Western thought.
Importantly for mathematics, 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 for entertainment. It was a
unifying element in his concept of Harmonia, the joining together of the cosmos
and the psyche. This may be why he is credited with discovering the link between
mathematical ratios and harmony. It is said that, while walking past a
55

blacksmith’s forge, 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 proportions, their resulting notes were
harmonic.
The hammers in the forge had individual weights of 6, 8, 9, and 12 units. Those
weighing 6 and 12 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 hammer of weight 6 was double that of the hammer
weighing 12, which corresponds with the ratio of their weights. The hammers of
weights 12 and 9 produced a harmonious sound—a perfect fourth—as their
weights were in the ratio 4:3. The notes made by the hammers of weights 12 and
8 were also harmonious—a perfect fifth—as their weights were in the ratio 3:2. In
contrast, the hammers of weights 9 and 8 were dissonant, as 9:8 is not a simple
mathematical ratio. By noticing that harmonious musical notes were connected to
numerical ratios, Pythagoras was the first to uncover the relationship between
mathematics and music.

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.

Creating a musical scale
56

Although scholars have questioned the story of the forge, Pythagoras is also
widely credited with another musical discovery. He is said to have experimented
with notes produced by lyre strings of different lengths. He found that while a
vibrating string produces a note with frequency f, halving the length of the string
produces a note an octave higher, with frequency 2f. When Pythagoras used the
same ratios that produced harmoniously sounding hammers, and applied them to
vibrating strings, he similarly produced notes in harmony with one another.
Pythagoras then constructed a musical scale, starting with one note and the note
an octave above it, filling in the notes between using perfect fifths.
This scale was used until the 1500s, when it was replaced by the even-tempered
scale, in which the notes between the two octaves are more evenly spaced.
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 different types of musical scales in use by different
cultures, the long tradition of Western music dates back to the Pythagoreans and
their quest to understand the relationship between music and mathematical
proportions.

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.

57

The legacy of Pythagoras
Pythagoras’s status as the most famous mathematician from antiquity is justified
by his contributions to geometry, number theory, and music. His ideas were not
always original, but the rigor with which he and his followers developed them,
using axioms and logic to build a system of mathematics, was a fine legacy for
those who succeeded him.
There is geometry in the humming of the strings, there is music in the spacing of the spheres.
Pythagoras

PYTHAGORAS
Pythagoras was born around 570 BCE on the Greek
island of Samos in the eastern Aegean Sea. His ideas
have influenced many of the greatest scholars in
history, from Plato to Nicolaus Copernicus, Johannes
Kepler, and Isaac Newton. Pythagoras is thought to
have traveled widely, assimilating ideas from scholars
in Egypt and elsewhere in the Middle East before
establishing his community of around 600 people in
Croton, southern Italy, around 518 BCE. This ascetic brotherhood required its
members to live for intellectual pursuits, while following strict rules of diet and
clothing. It is from this time onward that his theorem and other discoveries were
probably set down, although no records remain. At the age of 60, Pythagoras is
said to have married a young member of the community, Theano, and perhaps
had two or three children. Political upheaval in Croton led to a revolt against the
Pythagoreans. Pythagoras may have been killed when his school was set on fire,
or shortly afterward. He is said to have died around 495 BCE.
See also: Irrational numbers • The Platonic solids • Syllogistic logic • Calculating
pi • Trigonometry • The golden ratio • Projective geometry

58

IN CONTEXT
KEY FIGURE
Hippasus (5th century BCE)
FIELD
Number systems
BEFORE
19th century BCE Cuneiform inscriptions show that the Babylonians
constructed right-angled triangles and understood their properties.
6th century BCE In Greece, the relationship between the side lengths of a rightangled triangle is discovered, and is later attributed to Pythagoras.
AFTER
400 BCE Theodorus of Cyrene proves the irrationality of the square roots of the
nonsquare numbers between 3 and 17.
4th century BCE The Greek mathematician Eudoxus of Cnidus establishes a
strong mathematical foundation for irrational numbers.
Any number that can be expressed as a ratio of two integers—a fraction, a
decimal that either ends or repeats in a recurring pattern, or a percentage—is said
to be a rational number. All whole numbers are rational as they can be shown as
fractions divided by 1. Irrational numbers, however, cannot be expressed as a
ratio of two numbers
Hippasus, a Greek scholar, is believed to have first identified irrational numbers
in the 5th century BCE, as he worked on geometrical problems. He was familiar
with Pythagoras’s theorem, which states that the square of the hypotenuse in a
59

right-angled triangle is equal to the sum of the squares of the other two sides. He
applied the theorem to a right-angled triangle that has both shorter sides equal to
1. As 12 + 12 = 2, the length of the hypotenuse is the square root of 2.
Hippasus realized, however, that the square root of 2 could not be expressed as
the ratio of two whole numbers—that is, it could not be written as a fraction, as
there is no rational number that can be multiplied by itself to produce precisely 2.
This makes the square root of 2 an irrational number, and 2 itself is termed
nonsquare or square-free. The numbers 3, 5, 7, and many others are similarly
nonsquare and in each case, their square root is irrational. By contrast, numbers
such as 4 (22), 9 (32), and 16 (42) are square numbers, with square roots that are
also whole numbers and therefore rational.
The concept of irrational numbers was not readily accepted, although later Greek
and Indian mathematicians explored their properties. In the 9th century, Arab
scholars used them in algebra.

Hippasus may have encountered irrational numbers while exploring the relationship between
the length of the side of a pentagon and one side of a pentagram formed inside it. He found
that it was impossible to express it as a ratio between two whole numbers.

In decimal terms
60

The positional decimal system of Hindu–Arabic numeration allowed further study
of irrational numbers, which can be shown as an infinite series of digits after the
decimal point with no recurring pattern. For example, 0.1010010001… with an
extra zero between each successive pair of 1s, continuing indefinitely, is an
irrational number. Pi (π), which is the ratio of the circumference of a circle to its
diameter, is irrational. This was proved in 1761 by Johann Heinrich Lambert—
earlier estimations of π had been 3 or 22⁄7.
Between any two rational numbers, another rational number can always be
found. The average of the two numbers will also be rational, as will the average of
that number and either of the original numbers. Irrational numbers can also be
found between any two rational numbers. One method is to change a digit in a
recurring sequence. For example, an irrational number can be found between the
recurring numbers 0.124124… and 0.125125… by changing 1 to 3 in the second
cycle of 124, to give 0.124324…, and doing so again at the fifth, then ninth cycle,
increasing the gap between the replacement 3s by one cycle each time.
One of the great challenges of modern number theory has been establishing
whether there are more rational or irrational numbers. Set theory strongly
indicates that there are many more irrational numbers than rational numbers, even
though there are infinite numbers of each.

61

HIPPASUS
Details of Hippasus’s early life are sketchy, but it is
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 Pythagoras, Hippasus was a founder of a
Pythagorean sect called the Mathematici, which
fervently believed that all numbers were rational.
Hippasus is usually credited with discovering irrational numbers, an idea that
would have been considered heresy by the sect. According to one story,
Hippasus drowned when his fellow Pythagoreans threw him over the side of a
boat in disgust. Another story suggests that a fellow Pythagorean discovered
irrational numbers, but Hippasus was punished for telling the outside world
about them. The year of Hippasus’s death is not known but is likely to have
been in the 5th century BCE.
Key work
5th century BCE Mystic Discourse
See also: Positional numbers • Quadratic equations • Pythagoras • Imaginary and
complex numbers • Euler’s number

62

IN CONTEXT
KEY FIGURE
Zeno of Elea (c. 495–430 BCE)
FIELD
Logic
BEFORE
Early 5th century BCE The Greek philosopher Parmenides founds the Eleatic
school of philosophy in Elea, a Greek colony in southern Italy.
AFTER
350 BCE Aristotle produces his treatise Physics, in which he draws on the
concept of relative motion to refute Zeno’s paradoxes.
1914 British philosopher Bertrand Russell, who described Zeno’s paradoxes as
immeasurably subtle, states that motion is a function of position with respect to
time.

63

Zeno of Elea belonged to the Eleatic school of philosophy that flourished in
ancient Greece in the 5th century BCE. In contrast to the pluralists, who believed
that the Universe could be divided into its constituent atoms, Eleatics believed in
the indivisibility of all things.
Zeno wrote 40 paradoxes to show the absurdity of the pluralist view. Four of
these—the dichotomy paradox, Achilles and the tortoise, the arrow paradox, and
the stadium paradox—address motion. The dichotomy paradox shows the
absurdity of the pluralist view that motion can be divided. A body moving a
certain distance, it says, would have to reach the halfway point before it arrived at
the end, and in order to reach that halfway mark, it would first have to reach the
quarter-way mark, and so on ad infinitum. Because the body has to pass through
an infinite number of points, it would never reach its goal.
In the paradox of Achilles and the tortoise, Achilles, who is 100 times faster than
the tortoise, gives the creature a head start of 100 meters in a race. At the sound of
the starting signal, Achilles runs 100 meters to reach the tortoise’s starting point,
while the tortoise runs 1 meter, giving it a 1 meter lead. Undeterred, Achilles runs
another meter; however, in the same time, the tortoise runs one-hundredth of a
meter, 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 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
64

moving group can pass two people in the other moving group in a fixed time, but
only one person in the stationary group. The paradoxical conclusion is that half a
given time is equivalent to double that time.
Over the centuries, many mathematicians have refuted the paradoxes. The
development of calculus allowed mathematicians to deal with infinitesimal
quantities without resulting in contradiction.

The paradox of Achilles and the tortoise maintains that a fast object, such as Achilles, will
never catch up with a slow one, such as a tortoise. Achilles will get closer to the tortoise, but
never actually overtake it.

ZENO OF ELEA
Zeno of Elea was born around 495 BCE in the Greek city of Elea (now Velia, in
southern Italy). At a young age, he was adopted by the philosopher Parmenides,
and was said to have been “beloved” by him. Zeno was inducted into the school
of Eleatic thought, founded by Parmenides. At the age of around 40, Zeno
65

traveled to Athens, where he met Socrates. Zeno
introduced the Socratic philosophers to Eleatic ideas.
Zeno was renowned for his paradoxes, which
contributed to the development of mathematical rigor.
Aristotle later described him as the inventor of the
dialectical method (a method starting from two
opposing viewpoints) of logical argument. Zeno
collected his arguments in a book, but this did not survive. The paradoxes are
known from Aristotle’s treatise Physics, which lists nine of them.
Although little is known of Zeno’s life, the ancient Greek biographer Diogenes
claimed he was beaten to death for trying to overthrow the tyrant Nearchus. In a
clash with Nearchus, Zeno is reported to have bitten off the man’s ear.
See also: Pythagoras • Syllogistic logic • Calculus • Transfinite numbers • The
logic of mathematics • The infinite monkey theorem

66

IN CONTEXT
KEY FIGURE
Plato (c. 428–348 BCE)
FIELD
Geometry
BEFORE
6th century BCE Pythagoras identifies the tetrahedron, cube, and dodecahedron.
4th century BCE Theaetetus, an Athenian contemporary of Plato, discusses the
octahedron and icosahedron.
AFTER
c. 300 BCE Euclid’s Elements fully describes the five regular convex polyhedra.
1596 German astronomer Johannes Kepler proposes a model of the Solar
System, explaining it geometrically in terms of Platonic solids.
1735 Leonhard Euler devises a formula that links the faces, vertices, and edges
of polyhedra.

67

68

The perfect symmetry of the five Platonic solids was probably known to scholars
long before the Greek philosopher Plato popularized the forms in his dialogue
Timaeus, written in c. 360 BCE. Each of the five regular convex polyhedra—3-D
shapes with flat faces and straight edges—has its own set of identical polygonal
faces, the same number of faces meeting at each vertex, as well as equilateral
sides, and same-sized angles. Theorizing on the nature of the world, Plato
assigned four of the shapes to the classical elements: the cube (also known as a
regular hexahedron) was associated with earth; the icosahedron with water; the
octahedron with air; and the tetrahedron with fire. The 12-faced dodecahedron
was associated with the heavens and its constellations.

Composed of polygons
Only five regular polyhedra are possible—each one created either from identical
equilateral triangles, squares, or regular pentagons, as Euclid explained in Book
XIII of his Elements. To create a Platonic solid, a minimum of three identical
polygons must meet at a vertex, so the simplest is a tetrahedron— a pyramid
made up of four equilateral triangles. Octahedra and icosahedra are also formed
with equilateral triangles, while cubes are created from squares, and dodecahedra
are constructed with regular pentagons.
Platonic solids also display duality: the vertices of one polyhedron correspond to
the faces of another. For example, a cube, which has six faces and eight vertices,
and an octahedron (eight faces and six vertices) form a dual pair. A dodecahedron
(12 faces and 20 vertices), and an icosahedron (20 faces and 12 vertices) form
another dual pair. Tetrahedra, which have four faces and four vertices, are said to
be self-dual.

Shapes in the Universe?
69

Like Plato, later scholars sought Platonic solids in nature and the Universe. In
1596, Johannes Kepler reasoned that the positions of the six planets then known
(Mercury, Venus, Earth, Mars, Jupiter, and Saturn) could be explained in terms of
the Platonic solids. Kepler later acknowledged he was wrong, but his calculations
led him to discover that planets have elliptical orbits.
In 1735, Swiss mathematician Leonhard Euler noted a further property of
Platonic solids, later shown to be true for all polyhedra. The sum of the vertices
(V) minus the number of edges (E) plus the number of faces (F) always equals 2,
that is, V ˗ E + F = 2.
It is also now known that Platonic solids are indeed found in nature—in certain
crystals, viruses, gases, and the clustering of galaxies.

PLATO
Born around 428 BCE to wealthy Athenian parents,
Plato was a student of Socrates, who was also a family
friend. Socrates’ execution in 399 BCE deeply affected
Plato and he left Greece to travel. During this period
his discovery of the work of Pythagoras inspired a
love of mathematics. Returning to Athens, in 387 BCE
he founded the Academy, inscribing over its entrance
the words “Let no one ignorant of geometry enter
here.” Teaching mathematics as a branch of philosophy, Plato emphasized the
importance of geometry, believing that its forms—especially the five regular
convex polyhedra—could explain the properties of the Universe. Plato found
perfection in mathematical objects, believing they were the key to
understanding the differences between the real and the abstract. He died in
Athens around 348 BCE.
Key works
c. 375 BCE The Republic
c. 360 BCE Philebus
c. 360 BCE Timaeus
See also: Pythagoras • Euclid’s Elements • Conic sections • Trigonometry • NonEuclidean geometries • Topology • The Penrose tile
70

IN CONTEXT
KEY FIGURE
Aristotle (384–322 BCE)
FIELD
Logic
BEFORE
6th century BCE Pythagoras and his followers develop a systematic method of
proof for geometric theorems.
AFTER
c. 300 BCE Euclid’s Elements describes geometry in terms of logical deduction
from axioms.
1677 Gottfried Leibniz suggests a form of symbolic notation for logic,
anticipating the development of mathematical logic.
1854 George Boole publishes The Laws of Thought, his second book on
algebraic logic.
1884 The Foundations of Arithmetic by German mathematician Gottlob Frege
examines the logical principles underpinning mathematics.

71

In the Square of Opposition, S is a subject, such as “sugar,” and P a predicate, 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 be false); I and O are
subcontrary: both can be true but both cannot be false. I is 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.

In Classical Greece, there was no clear distinction between mathematics and
philosophy; the two were considered interdependent. For philosophers, one
important principle was the formulation of cogent arguments that followed a
logical progression of ideas. The principle was based on Socrates’ dialectal
method of questioning assumptions to expose inconsistencies and contradictions.
Aristotle, however, did not find this model entirely satisfactory, so he set about
determining a systematic structure for logical argument. First, he identified the
different kinds of proposition that can be used in logical arguments, and how they
can be combined to reach a logical conclusion. In Prior Analytics, he describes
the propositions as being of broadly four types, in the form of “all S are P,” “no S
are P,” “some S are P,” and “some S are not P,” where S is a subject, such as
sugar, and P the predicate—a quality, such as sweet. From just two such
propositions an argument can be constructed and a conclusion deduced. This is, in
essence, the logical form known as the syllogism: two premises leading to a
conclusion. Aristotle identified the structure of syllogisms that are logically valid,
those where the conclusion follows from the premises, and those that are not,
where the conclusion does not follow from the premises, providing a method for
both constructing and analyzing logical arguments.

72

Seeking a rigorous proof
Implicit in his discussion of valid syllogistic logic is the process of deduction,
working from a general rule in the major premise, such as “All men are mortal,”
and a particular case in the minor premise, such as “Aristotle is a man,” to reach a
conclusion that necessarily follows—in this case, “Aristotle is mortal.” This form
of deductive reasoning is the foundation of mathematical proofs.
Aristotle notes in Posterior Analytics that, even in a valid syllogistic argument, a
conclusion cannot be true unless it is based on premises accepted as true, such as
self-evident truths or axioms. With this idea, he established the principle of
axiomatic truths as the basis for a logical progression of ideas—the model for
mathematical theorems from Euclid onward.

ARISTOTLE
The son of a physician at the Macedonian court, Aristotle was born in 384 BCE,
in Stagira, Chalkidiki. At the age of about 17, he left to study at Plato’s
Academy in Athens, where he excelled. Soon after Plato’s death, antiMacedonian prejudice forced him to leave Athens. He continued his academic
73

work in Assos (now in Turkey). In 343 BCE, Philip II
recalled him to Macedonia to head the school at the
court; one of his students was Philip’s son, later
known as Alexander the Great.
In 335 BCE, Aristotle returned to Athens and founded
the Lyceum, a rival institution to the Academy. In 323
BCE, after Alexander’s death, Athens again became
fiercely anti-Macedonian, and Aristotle retired to his family estate in Chalcis, on
Euboea. He died there in 322 BCE.
Key works
c. 350 BCE Prior Analytics
c. 350 BCE Posterior Analytics
c. 350 BCE On Interpretation
335–323 BCE Nichomachean Ethics
335–323 BCE Politics
See also: Pythagoras • Zeno’s paradoxes of motion • Euclid’s Elements • Boolean
algebra • The logic of mathematics

74

IN CONTEXT
KEY FIGURE
Euclid (c. 300 BCE)
FIELD
Geometry
BEFORE
c. 600 BCE The Greek philosopher, mathematician, and astronomer Thales of
Miletus deduces that the angle inscribed inside a semicircle is a right angle. This
becomes Proposition 31 of Euclid’s Elements.
c. 440 BCE The Greek mathematician Hippocrates of Chios writes the first
systematically organized geometry textbook, Elements.
AFTER
c. 1820 Mathematicians such as Carl Friedrich Gauss, János Bolyai, and Nicolai
Ivanovich Lobachevsky begin to move toward hyperbolic non-Euclidean
geometry.
Euclid’s Elements has a strong claim for being the most influential mathematical
work of all time. It dominated human conceptions of space and number for more
than 2,000 years and was the standard geometrical textbook until the start of the
1900s.
Euclid lived in Alexandria, Egypt, in around 300 BCE, when the city was part of
the culturally rich Greek-speaking Hellenistic world that flourished around the
Mediterranean Sea. He would have written on papyrus, which is not very durable;

75

all that remains of his work are the copies, translations, and commentaries made
by later scholars.
There is no royal road to geometry.
Euclid

Collection of works
The Elements is a collection of 13 books that range widely in subject matter.
Books I to IV tackle plane geometry—the study of flat surfaces. Book V
addresses the idea of ratio and proportion, inspired by the thinking of the Greek
mathematician and astronomer Eudoxus of Cnidus. Book VI contains more
advanced plane geometry. Books VII to IX are devoted to number theory and
discuss the properties and relationships of numbers. The long and difficult Book
X deals with incommensurables. Now known as irrational numbers, these
numbers cannot be expressed as a ratio of integers. Books XI to XIII examine
three-dimensional solid geometry.
Book XIII of the Elements is actually attributed to another author—Athenian
mathematician and disciple of Plato, Theaetetus, who died in 369 BCE. It covers
the five regular convex solids—the tetrahedron, cube, octahedron, dodecahedron,
and icosahedron, which are often called the Platonic solids—and is the first
recorded example of a classification theorem (one that itemizes all possible
figures given certain limitations).
Euclid is known to have written an account of conic sections, but this work has
not survived. Conic sections are figures formed from the intersection of a plane
and a cone and they may be circular, elliptical, or parabolic in shape.

EUCLID
Details of Euclid’s date and place of birth are
unknown and knowledge of his life is scant. It is
thought that he studied at the Academy in Athens,
which had been founded by Plato. In the 5th century
CE, the Greek philosopher Proclus wrote in his history
of mathematicians that Euclid taught at Alexandria
during the reign of Ptolemy I Soter (323–285 BCE).

76

Euclid’s work covers two areas: elementary geometry and general
mathematics. In addition to the Elements, he wrote about perspective, conic
sections, spherical geometry, mathematical astronomy, number theory, and the
importance of mathematical rigor. Several of the works attributed to Euclid
have been lost, but at least five have survived to the 21st century. It is thought
that Euclid died between the mid-4th century and the mid-3rd century BCE.
Key works
Elements
Conics
Catoptrics
Phaenomena
Optics

World of proof
The title of Euclid’s work has a particular meaning that reflects his mathematical
approach. In the 1900s, British mathematician John Fauvel maintained that the
meaning of the Greek word for “element,” stoicheia, changed over time, from “a
constituent of a line,” such as an olive tree in a line of trees, to “a proposition
used to prove another,” and eventually evolved to mean “a starting point for many
other theorems.” This is the sense in which Euclid used it. In the 5th century CE,
the philosopher Proclus talked of an element as “a letter of an alphabet,” with
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combinations of letters creating words in the same way that combinations of
axioms—statements that are self-evidently true—create propositions.

This opening page of Euclid’s Elements shows illuminated Latin text with diagrams and
comes from the first printed edition, produced in Venice in 1482.

Logical deductions
Euclid was not writing in a vacuum; he built upon foundations laid by a number
of influential Greek mathematicians who came before him. Thales of Miletus,
Hippocrates, and Plato (among others) had all begun to move toward the
mathematical mindset that Euclid so brilliantly formalized: the world of proof. It
is this that makes Euclid unique; his writings are the earliest surviving example of
fully axiomatized mathematics. He identified certain basic facts and progressed
from there to statements that were sound logical deductions (propositions). Euclid
also managed to assemble all the mathematical knowledge of his day, and
organize it into a mathematical structure where the logical relationships between
the various propositions were carefully explained.

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Euclid faced a Herculean task when he attempted to systematize the mathematics
that lay before him. In devising his axiomatic system, he began with 23
definitions for terms such as point, line, surface, circle, and diameter. He 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 straight line
segment, a circle can be drawn having the segment as its radius and one endpoint
as its center; all right angles are equal to one another; and a postulate about
parallel lines (see Euclid’s five postulates).
He then went on to add five axioms, or common notions; if A = B and B = C,
then A = C; if A = B and C = D, A + C = B + D; if A = B and C = D, then A - C =
B - D; if A coincides with B, then A and B are equal; and the whole of A is
greater than part of A.
To prove Proposition 1, Euclid drew a line with endpoints labeled A and B.
Taking each endpoint as a center, he then drew two intersecting circles, so that
each had the radius AB. This used his third postulate. Where the circles met, he
called that point C, and he could draw two more lines AC and BC, calling on his
first postulate. The radius of the two circles is the same, so AC = AB and BC =
AB; this means that AC = BC, which is Euclid’s first axiom (things that are equal
to the same thing are also equal to one another). It follows that AB = BC = CA,
meaning that he had drawn an equilateral triangle on AB.
In Latin translations of Elements, deductions end with the letters QEF (quod erat
faciendum, meaning “which was to be [and has been] done.” Logical proofs end
with QED (quod erat demonstrandum, meaning “which was to be [and has been]
demonstrated”).
The equilateral triangle construction is a good example of Euclid’s method. Each
step has to be justified by reference to the definitions, the postulates, and the
axioms. Nothing else can be taken as obvious, and intuition is regarded as
potentially suspect.
Euclid’s very first proposition was criticized by later writers. They noted, for
instance, that Euclid did not justify or explain the existence of C, the point of
intersection of the two circles. Although apparent, it is not mentioned in his
preliminary assumptions. Postulate 5 talks about a point of intersection, but that is
between two lines, and not two circles. Similarly, one of the definitions describes
a triangle as a plane figure bounded by three lines, which all lie in that plane.

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However, it seems that Euclid did not explicitly show that the lines AB, BC, and
CA lie in the same plane.
Postulate 5 is also known as the “parallel postulate” because it can be used to
prove properties of parallel lines. It says that if a straight line crossing two straight
lines (A, B) creates interior angles on one side that total less than two right angles
(180°), lines A and B will eventually cross on that side, if extended indefinitely.
Euclid did not use it until Proposition 29, in which he stated that one condition for
a straight line crossing two parallel lines was that the interior angles on the same
side were equal to two right angles. The fifth postulate is more elaborate than the
other four, and Euclid himself seems to have been wary of it.
A vital part of any axiomatic system is to have enough axioms, and postulates in
the case of Euclid, to derive every true proposition, but to avoid superfluous
axioms that can be derived from others. Some asked whether the parallel postulate
could be proved as a proposition using Euclid’s common notions, definitions, and
the other four postulates; if it could, the fifth was unnecessary. Euclid’s
contemporaries and later scholars made unsuccessful attempts to construct such a
proof. Finally, in the 1800s, the fifth postulate was ruled both necessary for
Euclid’s geometry and independent of his other four postulates.

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To construct an equilateral triangle, for Proposition 1, Euclid drew a line and centered a
circle on its endpoints, here A and B. By drawing a line from each endpoint to C, where the
circles intersect, he created a triangle with sides AB, AC, and BC of equal length.
Geometry is knowledge of what always exists.
Plato

Beyond Euclidean geometry
The Elements also examines spherical geometry, an area explored by two of
Euclid’s successors, Theodosius of Bithynia and Menelaus of Alexandria. While
Euclid’s definition of “a point” addresses a point on the plane, a point can also be
understood as a point on a sphere.
This raises the question of how Euclid’s five postulates can be applied to the
sphere. In spherical geometry, almost all the axioms look different from the
postulates set out in Euclid’s Elements. The Elements gave rise to what is called
Euclidean geometry; spherical geometry is the first example of a non-Euclidean
geometry. The parallel postulate is not true for spherical geometry, where all pairs
of lines have points in common, nor for hyperbolic geometry, where they can
meet infinite numbers of times.

The first 16 propositions in Book 1
Proposition 1

On a given finite straight line, to construct an equilateral triangle.

Proposition 2

To place at a given point (as an extremity) a straight line equal to a
given straight line.

Proposition 3

Given two unequal straight lines, to cut off from the greater a straight

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line equal to the less.

Proposition 4

If two sides of one triangle are equal in length to two sides of another
triangle, and if the angles contained by each pair of equal sides are
equal, then the base of one triangle will equal 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 in the other triangle.

Proposition 5

In an isosceles triangle, the angles at the base are equal to one another,
and, if the equal straight lines are extended below the base, the angles
under the base will also be equal to one another.

Proposition 6

If in a triangle two angles are equal to one another, the sides separated
from the third side by these angles will also be equal.

Proposition 7

Given two straight lines constructed on a straight line (from its
extremities) and meeting in a point, t