Aulos Kapreilios Timotheos, Slave Trader by M.I. Finley (1968)

This blog post started out as simple notes on a short essay by the noted historian of the ancient world, Sir Moses Finley (1912 to 1986) – until I stumbled on the wider context of the essay on the internet, which I then try to summarise.

Aulos Kapreilios Timotheos, Slave Trader

The essay ‘Aulos Kapreilios Timotheos, Slave Trader’ was published in the early 1960s, then included in a slim Pelican paperback collection, ‘Aspects of Antiquity’, published in 1968, which I picked up sometime in the 1980s.

It is far from being a big definitive essay on the huge subject of slavery in antiquity. Rather, it’s a set of meditations which flow from contemplating just one artifact from the ancient world, a seven-feet-high, finely decorated marble tombstone to this man, Aulos Kapreilios Timotheos.

Tombstone of Aulos Kapreilios Timotheos, Slave Trader

The tombstone

This tombstone was found at a town near the border between modern Turkey and Greece. It shows three carved scenes: a typical banquet at the top; a work scene in the middle; and on the bottom, a depiction of 8 slaves, chained together by the neck, being led in single file, accompanied by two women and two children, not chained, preceded by a man who is obviously in charge. Between the top and second row is an inscription in Greek, reading:

Aulos Kapreilios Timotheos, freedman of Aulos, slave trader

Apparently what makes this stone rare and unusual is its blunt candour. In the scattered writings we have from the ancient world slave trading was looked down on, sometimes despised, which is odd because the entire economies of ancient Greece and Rome relied on slaves in enormous numbers. But clearly, the writing classes – the people who left opinions for us to read – were ambivalent about it at best.

The American South

Finley compares and contrasts the situation in the ancient world with that in the Southern United States in the nineteenth century. American slave owners were uneasily aware that the rest of the civilised world had abolished slavery and strongly disapproved of them. Hence their increasingly anxious over-compensating justification of the ‘peculiar institution’.

The ancient Greeks and Romans had no external voice of conscience to upbraid them. The reverse. Everywhere they looked they saw all other societies of their time practising slavery.

The racial justification for slavery

The slave society of the Deep South justified its exploitation with widespread propaganda about the intrinsic inferiority of black people. You don’t read far in any text about the American civil war without coming across southern ideologues using the Bible or any other spurious means they can lay hands on to justify the intrinsic superiority of whites and the intrinsic inferiority of blacks. Plenty of authors and politicians claimed that blacks could only find true happiness in the condition of slavery, blacks are children who need the strong hand of a father etc etc.

So a black person in America could never lose the stigma associated with slavery, even if they were free, even if they lived in the north, ran a business, lived a free life, could never be completely free.

The raceless basis of ancient slavery

The situation was drastically different in the ancient world because slavery wasn’t associated with any particular race or ethnicity. Literally anyone could be enslaved – in Spain, in Gaul, in Greece itself, conquering Roman armies enslaved entire cities of white Caucasians.

The crucial point is that there were no specifically slave races or nationalities. Literally anyone and everyone might be enslaved, and which groups predominated at one time or another depended on politics and war. (p.157)

The association of slavery with skin colour was an invention of the Atlantic slave trade of the 17th and 18th centuries.

Freed names

Back to Aulos – his first two names, Aulos Kapreilio were those of his master, which he took when he was made free, as per Roman custom. Timotheos was his slave name.

Roman slave names

In the early days of slavery Romans gave their slaves names like Marcipor or Lucipor which was simply a contraction of Marcus puer or Lucius puer, puer being Latin for ‘boy’ (hence the English word ‘puerile’, which has come to mean ‘childishly silly and immature’).

From the year of the twin defeats of Carthage and Corinth, 146 BC, the number of slaves began to steadily increase and so they needed more names.

After 146 the empire became unofficially divided into a Latin-speaking West and a Greek-speaking East, and so slave names sometimes indicate a slave’s origins, east or west.


A city on the south coast of Anatolia, became notorious as a slave market. But maybe the epicentre of the ancient slave trade was the island of Delos

The people of Phrygia in central were notorious for selling their own children into captivity. Many slaves from Scythia (the area to the north of the Black Sea) were bought from their own chieftains, captives in their own wars, or children, or simply human levies, like tax, sold at a profit (p.163).

Slave sales

Given the millions of men, women and children who were slaves it is notable that we have just two visual depictions of an actual slave auction. In both of them a male slave stands on a platform while another man, presumably the buyer, lifts his tunic to admire his strong thighs.

The condition of a slave

is to be brought into a new and alien society violently and traumatically; to be torn not only from his homeland but from all the relationships which provide identity and psychological stability, with family, kin, tribe, village, region, gods, customs, dress – everything.

All this is replaced with just one cardinal relationship – with the slave’s male owner who controls not only every aspect of his physical existence, but his mental horizons, the language he has to use, the new religion he has to practice, rules he has to obey – everything.

Slave sexual exploitation

Complete control over the person of slaves meant the master class had unfettered unlimited sexual access to all slaves, male, female, young or old. As I’ve read the chatty odes of Horace or elegiacs of Tibullus, Propertius or Ovid, I have been disturbed again and again by the casual way they talk about being ‘given’ slaves (of either gender) for sexual purposes.

Slave punishments

The most chilling thing for me, though, has been the casual references, in all the Roman literature I’ve read from Plautus onwards, of the horrific punishments slaves could be subject to, starting with whipping and escalating through torture, having limbs deliberately broken, and so on, up to the ultimate punishment of crucifixion.

Finley returns to the attempts of Americans to justify slavery through the intrinsic inferiority of one race and say not only was it not attempted in the ancient world, it was actively disproved by the case of the Greeks.

Greek revenge

After the brutal conquest of the Greek League in 146 BC, over the next few centuries hundreds of thousands of Greek men, women and children were brought back to Italy as slaves. However, in the long term this caused a kind of cultural revolution. The Gauls or Germans might have been considered ‘barbarians’ (they wore trousers, for God’s sake!) but the Greeks were citizens of the culture which had taught the Romans literature, philosophy and architecture. Hard to maintain the fiction that these people were in any way ‘inferior’. On the contrary many of them, while remaining technically ‘slaves’, rose to become secretaries, assistants or teachers to the master’s children.


This leads into another important issues, which is manumission, which is the fancy word for freeing your slaves. The Romans became famous among the cultures of the ancient world for freeing their slaves, as reward for loyal service. It was a disconcertingly simple procedure – the owner declaring the slave free, maybe touching them or gently pushing them away, and a state official such as a consul or a praetor touching the slave with a rod called a vindicta and pronouncing him or her to be free.

The slave’s head was shaved and a pileus was placed upon it. The pileus was a brimless felt cap of undyed wool. Based on what we can see in surviving frescos, sculptures, and coins, the pileus ranged from a short cone to a gumdrop shape. It was the identifying garment of a freedman.

Anyway, we know that the rate of manumission became a real problem in Roman society because the emperor Augustus passed laws trying to limit it:

He established maxima on a sliding scale, according to which no one man was allowed to free more than one hundred slaves in his will. (p.158)

Finley points out a notorious contradiction in Roman attitudes to slavery: which is that noted jurists such as Florentinus clearly stated that slavery as an institution was ‘contrary to nature’, that this idea was shared in some of the literature and incorporated into legal codes – and yet it didn’t make any difference to the actual practice.

He instances the moral philosopher Seneca who freely admitted that a slave is a person with a soul like you and me, but from this premise he draws the conclusion that one should live on friendly terms with one’s slaves, dine with them, converse with them etc – everything except actually free them, which seems beyond the scope of his philosophy (p.164).


Because, as Finley points out, war was central to the entire institution of slavery and the slave trade.

The ancient world was one of unceasing warfare, and the accepted rule was that the victor had absolute rights over the person and property of the captives, without distinction between soldiers and civilians. (p.159)


went to Gaul an impoverished nobleman and returned a multi-millionaire and this was partly because of the huge number of captives he seized and sold into slavery, taking a commission. After he captured the town of the Atuatuci he sold the entire population of 53,000 into lifelong slavery. After the Battle of Alesia in 52 BC he gave one captive to every one of his legionaries.

War slavers

Enormous numbers like this would slow an army down so by Caesar’s time arrangements were in place to have slave traders accompany the army, or meet them at arranged rendezvous, there to buy the newly captured slaves, take them off the commander’s hands, and do with them as he please, tramp them all the way back to Italy or sell them locally.

Maybe the procession on Timotheos’s tombstone depicts such a merchant marching off some of his new merchandise.


From a business point of view the problem was the extreme unpredictability of war. Hence the inexorable rise from 150 or so onwards of piracy in the Mediterranean. This wasn’t a case of a few swashbuckling privateers but ‘a complex business network of pirates, kidnappers and slave dealers’, with its headquarters at Side and its main emporium on the Greek island of Delos. Finley quotes the figure I’ve read elsewhere that the docks and warehouses of Delos were extended so that at its peak it turned over as many as 10,000 slaves a day.

(On the subject of scale, Finley says that as early as the 4th century BC the number of slaves working in Athens’s silver mines was probably as high as 30,000.)


The rise and rise of slavery went hand in hand with a crucial socio-economic development in mainland Italy. This was the eradication of the small family farm – the kind of place which Virgil and Horace idolised as the cradle of morality and right living – and its replacement by vast estates or latifundia owned by enormously rich absentee landlords and worked by slave gangs often working in chains.

The servile wars

The scale of the exploitation and the resentment it bred led to the three major slave revolts which escalated so far as to be called ‘wars’, the so-called Servile Wars:

  • First Servile War (135 to 132 BC) in Sicily, led by Eunus, a former slave claiming to be a prophet, and Cleon from Cilicia
  • Second Servile War (104 to 100 BC) in Sicily, led by Athenion and Tryphon
  • Third Servile War (73 to 71 BC) on mainland Italy, led by Spartacus


Specialist skills were in great demand. If a slave could play music, recite poetry, take dictation or any number of other skills then he or she might secure a relatively comfortable lifestyle. Alternatively, slaves could be trained, specially if started young.

Many slaves became masters of crafts and trades; the chain-ganged brute labour of the countryside was matched by highly skilled slaves in more urban settings who worked in potteries or textile mills, on temples and other public works, sometimes performing artistic and delicate work.

The sheer number of slaves present at every level of Roman society, participating in a huge range of activities, suggests the ‘condition’ or psychology of slavery must have been hugely varied, as varied, maybe, as the number of individual slaves.

The end of ancient slavery

Slavery ended not because of any abolitionist movement but because of profound socio-economic changes in the Roman Empire. These slow economic transformations replaced both the ‘chattel slave’ and the free peasant of Virgil and Horace’s dreams, with a new social class, a new type of ‘bondsman’ – the colonus, the adscripticius, who was himself to evolve into the serf.

For the most part. But slavery didn’t disappear from Europe, not even from the Empire. Finley tells us that when the sixth-century emperor Justinian drew up a codification of all existing laws, the issues thrown up by slavery took up more space than any other topic.

The essay in the context of Finley’s career

Online you can read the first page of an essay about Finley and slavery by the American academic, Arnaldo Momigliano. This tells us that Finley had a lifelong interest in the question of slavery in the ancient world and that the present essay repeats some themes and ideas already discussed in his 1958 essay, ‘Was Greek civilisation based on slave labour?’ (itself included in a 1960 collection, ‘Slavery and Classical Antiquity’) and takes its place alongside other papers on the subject gathered in the 1981 volume, ‘Economy and Society in Ancient Greece’.

Apparently, Finley’s ideas about slavery were most fully expressed in the book-length study, ‘Ancient Slavery and Modern Ideology’, published in 1980 (so when he was 68). If you go looking for it on Amazon, you find the latest imprint of the book and discover that it was republished in 1998 with new material by an academic named Brent Shaw.

This volume, ‘Ancient Slavery and Modern Ideology’, isn’t a history of slavery as such, it’s an account of the interpretations succeeding ages have made of slavery in the ancient world, according to each era’s ideologies and principles. In what follows I’m indebted to the excellent review of ‘Ancient Slavery and Modern Ideology’ on Amazon by Richard Mathisen. To be clear, I’m putting Mathisen’s words in italics.

Richard Mathisen’s summary of ‘Ancient Slavery and Modern Ideology’

For Finley, there have been only five genuine slave societies, two ancient (Greece and Rome), and three modern (the Caribbean, Brazil, and the American South).

Historians of ancient societies have always been affected by ideological bias. Classical historians admired Greek and Roman civilisations so they downplayed the ugly aspects of slavery. Christian historians tried to claim that Christianity ended slavery, but it didn’t. Marxist historians wanted to interpret ancient slavery through their lens of class war while anti-Marxist historians took the opposite view.

While ancient slavery had no racial component, modern historians are influenced by racial concerns so that every “new interpretation of slavery has professed to be more anti-racist than the one it replaces.”

Finley’s aim is to trace the distorting effect of each of these ideologies on the history of slavery. Finley explains the emergence of ancient slave societies, which requires three conditions: private ownership of land, commodified systems of production, and a shortage of labour. He considers societal attitudes toward the humanity of slaves and traces the end of slavery as it transitioned into feudalism.

Finley carefully defines slavery, because many examples of forced labour have existed, including Egyptian pyramids, Assyrian and Babylonian empires, Spartan helots, feudal serfs, and indentured servants, but they were not slaves. Indeed, he notes that the most unusual labour system in history is modern free wage labour, with individuals free to move.

This leads to Finley’s real interest. What factors led to ancient slavery? When did it start, when did it end, and why? What aspects of ancient society were part of slavery’s support system? What were the ideological presuppositions of the Greeks and Romans? Why was the legitimacy of slavery never questioned in ancient times, even during slave revolts? Why did slavery exist only in certain areas of Rome, such as Italy and Sicily? Could slavery ever come back again in the modern world, if the necessary conditions seemed to demand it?

When re-issuing Finley’s book, Brent Shaw added a 1981 response by Finley to his critics and a 1979 essay on “Slavery and the Historians.” Shaw himself wrote a 76-page essay updating the slavery debate since 1980.

The vast historiography of a complex subject

All this builds up to quite a complex picture which can be summarised as:

  • during his career Finley wrote a number of essays about slavery in the ancient world
  • his main statement on the subject is a book which describes the changing interpretations of ancient slavery made by the leading ideologies of different eras
  • critics criticised this book
  • Finley wrote an essay addressing these criticisms
  • Brent Shaw added a long essay updating the debate since 1980 (presumably up till 1998, when this new edition was published)

But quite obviously a lot of this is very old. When I skimmed through the passages of ‘Ancient Slavery and Modern Ideology’ available on Amazon, I caught references to the Soviet Union. The idea of describing an aspect of the ancient world as it has been interpreted, reinterpreted and misinterpreted by the leading ideologies of successive ages sounds really interesting, but…1980. Surely I ought to be reading something far more up to date.

And then, when I saw that the Arnaldo Momigliano essay about Finley had been published in a periodical titled ‘Slavery and Abolition: A Journal of Slave and Post-Slave Studies’, my heart sank. Every month or so since the late 1970s this journal has been publishing articles about slavery. By now there must be a mountain of content – and I bet there are other journals on the subject, not to mention the hundreds of thousands of academic papers and tens of thousands of books, and hundreds of conferences which must have been held on the subject. How long would it take to read all the relevant studies, paper and books on the subject? A year? Three years? I’d like to learn and understand more but do I have the time required? Does anyone have the time?


‘Aulos Kapreilios Timotheos, Slave Trader’ was included in a collection of essays by M.I. Finley titled ‘Aspects of Antiquity’, published by Penguin books in 1968. References are to the 1977 Penguin paperback edition.

Related link

Roman reviews

Alex’s Adventures In Numberland by Alex Bellos (2010)

Alexander Bellos (born in 1969) is a British writer and broadcaster. He is the author of books about Brazil and mathematics, as well as having a column in The Guardian newspaper. After adventures in Brazil (see his Wikipedia page) he returned to England in 2007 and wrote this, his first book. It spent four months in the Sunday Times bestseller list and led on to five more popular maths books.

It’s a hugely enjoyable read for three reasons:

  1. Bellos immediately establishes a candid, open, good bloke persona, sharing stories from his early job as a reporter on the Brighton Argus, telling some colourful anecdotes about his time in Brazil and then being surprisingly open about the way that, when he moved back to Britain, he had no idea what to do. The tone of the book is immediately modern, accessible and friendly.
  2. However this doesn’t mean he is verbose. The opposite. The book is packed with fascinating information. Every single paragraph, almost every sentence contains a fact or insight which makes you sit up and marvel. It is stufffed with good things.
  3. Lastly, although its central theme is mathematics, it approaches this through a wealth of information from the humanities. There is as much history and psychology and anthropology and cultural studies and philosophy as there is actual maths, and these are all subjects which the average humanities graduate can immediately relate to and assimilate.

Chapter Zero – A Head for Numbers

Alex meets Pierre Pica, a linguist who’s studied the Munduruku people of the Amazon and discovered they have little or no sense of numbers. They only have names for numbers up to five. Also, they cluster numbers together logarithmically i.e. the higher the number, the closer together they clustered them. Same thing is done by kindergarten children who only slowly learn that numbers are evenly spaced, in a linear way.

This may be because small children and the Munduruku don’t count so much as estimate using the ratios between numbers.

It may also be because above a certain number (five) Stone Age man needed to make quick estimates along the lines of, Are there more wild animals / members of the other gang, than us?

Another possibility is that distance appears to us to be logarithmic due to perspective: the first fifty yards we see in close detail, the next fifty yards not so detailed, beyond 100 yards looking smaller, and so on.

It appears that we have to be actively taught when young to overcome our logarithmic instincts, and to apply the rule that each successive whole number is an equal distance from its predecessor and successor i.e. the rational numbers lies along a straight line at regular intervals.

More proof that the logarithmic approach is the deep, hard-wired one is the way most of us revert to its perspective when considering big numbers. As John Allen Paulos laments, people make no end of fuss about discrepancies between 2 or 3 or 4 – but are often merrily oblivious to the difference between a million or a billion, let alone a trillion. For most of us these numbers are just ‘big’.

He goes on to describe experiments done on chimpanzees, monkeys and lions which appear to show that animals have the ability to estimate numbers. And then onto experiments with small babies which appear to show that as soon as they can focus on the outside world, babies can detect changes in number of objects.

And it appears that we also have a further number skill, that guesstimating things – the journey takes 30 or 40 minutes, there were twenty or thirty people at the party, you get a hundred, maybe hundred and fifty peas in a sack. When it comes to these figures almost all of us give rough estimates.

To summarise:

  • we are sensitive to small numbers, acutely so of 1, 2, 3, 4, less so of 5, 6, 7, 8, 9
  • left to our own devices we think logarithmically about larger numbers i.e lose the sense of distinction between them, clump them together
  • we have a good ability to guesstimate medium size numbers – 30, 40, 100

But it was only with the invention of notation, a way of writing numbers down, that we were able to create the linear system of counting (where every number is 1 larger than its predecessor, laid out in a straight line, at regular intervals).

And that this cultural invention enabled human beings to transcend our vague guesstimating abilities, and laid the basis for the systematic manipulation of the world which followed

Chapter One – The Counter Culture

The probable origins of counting lie in stock taking in the early agricultural revolution some 8,000 years ago.

We nowadays count using a number base 10 i.e. the decimal system. But other bases have their virtues, especially base 12. It has more factors i.e. is easier to divide: 12 can be divided neatly by 2, 3, 4 and 6. A quarter of 10 is 2.5 but of 12 is 3. A third of 10 is 3.333 but of 12 is 4. Striking that a version of the duodecimal system (pounds, shillings and pence) hung on in Britain till we finally went metric in the 1970s. There is even a Duodecimal Society of America which still actively campaigns for the superiority of a base 12 counting scheme.

Bellos describes a bewildering variety of other counting systems and bases. In 1716 King Charles XII of Sweden asked Emmanuel Swedenborg to devise a new counting system with a base of 64. The Arara in the Amazon count in pairs, the Renaissance author Luca Paccioli was just one of hundreds who have devised finger-based systems of counting – indeed, the widespread use of base 10 probably stems from the fact that we have ten fingers and toes.

He describes a complicated Chinese system where every part of the hand and fingers has a value which allows you to count up to nearly a billion – on one hand!

The Yupno system which attributes a different value for parts of the body up to its highest number, 33, represented by the penis.

Diagram showing numbers attributed to parts of the body by the Yupno tribe

Diagram showing numbers attributed to parts of the body by the Yupno tribe

There’s another point to make about his whole approach which comes out if we compare him with the popular maths books by John Allen Paulos which I’ve just read.

Paulos clearly sees the need to leaven his explanations of comparative probability and Arrow’s Theorem and so on with lighter material and so his strategy is to chuck into his text things which interest him: corny jokes, anecdotes about baseball, casual random digressions which occur to him in mid-flow. But al his examples clearly 1. emanate from Paulos’s own interests and hobby horses (especially baseball) and 2. they are tacked onto the subjects being discussed.

Bellos, also, has grasped that the general reader needs to be spoonfed maths via generous helpings of other, more easily digestible material. But Bellos’s choice of material arises naturally from the topic under discussion. The humour emerges naturally and easily from the subject matter instead of being tacked on in the form of bad jokes.

You feel yourself in the hands of a master storyteller who has all sorts of wonderful things to explain to you.

In fourth millennium BC, an early counting system was created by pressing a reed into soft clay. By 2700 BC the Sumerians were using cuneiform. And they had number symbols for 1, 10, 60 and 3,600 – a mix of decimal and sexagesimal systems.

Why the Sumerians grouped their numbers in 60s has been described as one of the greatest unresolved mysteries in the history of arithmetic. (p.58)

Measuring in 60s was inherited by the Babylonians, the Egyptians and the Greeks and is why we still measure hours in 60 minutes and the divisions of a circle by 360 degrees.

I didn’t know that after the French Revolution, when the National Convention introduced the decimal system of weights and measures, it also tried to decimalise time, introducing a new system whereby every day would be divided into ten hours, each of a hundred minutes, each divided into 100 seconds. Thus there were a very neat 10 x 100 x 100 = 100,000 seconds in a day. But it failed. An hour of 60 minutes turns out to be a deeply useful division of time, intuitively measurable, and a reasonable amount of time to spend on tasks. The reform was quietly dropped after six months, although revolutionary decimal clocks still exist.

Studies consistently show that Chinese children find it easier to count than European children. This may be because of our system of notation, or the structure of number names. Instead of eleven or twelve, Chinese, Japanese and Koreans say the equivalent of ten one, ten two. 21 and 22 become two ten one and two ten two. It has been shown that this makes it a lot simpler and more intuitive to do basic addition and subtraction.

Bellos goes on to describe the various systems of abacuses which have developed in different cultures, before explaining the phenomenal popularity of abacus counting, abacus clubs, and abacus championships in Japan which helps kids develop the ability to perform anzan, using the mental image of an abacus to help its practitioners to sums at phenomenal speed.

Chapter Two – Behold!

The mystical sense of the deep meaning of numbers, from Pythagoras with his vegetarian religious cult of numbers in 4th century BC Athens to Jerome Carter who advises leading rap stars about the numerological significance of their names.

Euclid and the elegant and pure way he deduced mathematical theorems from a handful of basic axioms.

A description of the basic Platonic shapes leads into the nature of tessalating tiles, and the Arab pioneering of abstract design. The complex designs of the Sierpinski carpet and the Menger sponge. And then the complex and sophisticated world of origami, which has its traditionalists, its pioneers and surprising applications to various fields of advanced science, introducing us to the American guru of modern origami, Robert Lang, and the Japanese rebel, Kazuo Haga, father of Haga’s Theorem.

Chapter Three – Something About Nothing

A bombardment of information about the counting systems of ancient Hindus, Buddhists, about number symbols in Sanskrit, Hebrew, Greek and Latin. How the concept of zero was slowly evolved in India and moved to the Muslim world with the result that the symbols we use nowadays are known as the Arabic numerals.

A digression into ‘a set of arithmetical tricks known as Vedic Mathematics ‘ devised by a young Indian swami at the start of the twentieth century, Bharati Krishna Tirthaji, based on a series of 16 aphorisms which he found in the ancient holy texts known as the Vedas.

Shankaracharya is a commonly used title of heads of monasteries called mathas in the Advaita Vedanta tradition. Tirthaji was the Shankaracharya of the monastery at Puri. Bellos goes to visit the current Shankaracharya who explains the closeness, in fact the identity, of mathematics and Hindu spirituality.

Chapter Four – Life of Pi

An entire chapter about pi which turns out not only to be a fundamental aspect of calculating radiuses and diameters and volumes of circles and cubes, but also to have a long history of mathematicians vying with each other to work out its value to as many decimal places as possible (we currently know the value of pi to 2.7 trillion decimal places) and the surprising history of people who have set records reciting the value if pi.

Thus, in 2006, retired Japanese engineer Akira Haraguchi set a world record for reciting the value of pi to the first 100,000 decimal places from memory! It took 16 hours with five minute beaks every two hours to eat rice balls and drink some water.

There are several types or classes of numbers:

  • natural numbers – 1, 2, 3, 4, 5, 6, 7…
  • integers – all the natural numbers, but including the negative ones as well – …-3, -2, -1, 0, 1, 2, 3…
  • fractions
  • which are also called rational numbers
  • numbers which cannot be written as fractions are called irrational numbers
  • transcendent numbers – ‘a transcendental number is an irrational number that cannot be described by an equation with a finite number of terms’

The qualities of the heptagonal 50p coin and the related qualities of the Reuleux triangle.

Chapter Five – The x-factor

The origin of algebra (in Arab mathematicians).

Bellos makes the big historical point that for the Greeks (Pythagoras, Plato, Euclid) maths was geometric. They thought of maths as being about shapes – circles, triangles, squares and so on. These shapes had hidden properties which maths revealed, thus giving – the Pythagoreans thought – insight into the secret deeper values of the world.

It is only with the introduction of algebra in the 17th century (Bellos attributes its widespread adoption to Descartes’s Method in the 1640s) that it is possible to fly free of shapes into whole new worlds of abstract numbers and formulae.

Logarithms turn the difficult operation of multiplication into the simpler operation of addition. If X x Y = Z, then log X + log Y = log Z. They were invented by a Scottish laird John Napier, and publicised in a huge book of logarithmic tables published in 1614. Englishman Henry Briggs established logarithms to base 10 in 1628. In 1620 Englishman Edmund Gunter marked logarithms on a ruler. Later in the 1620s Englishman William Oughtred placed two logarithmic rulers next to each other to create the slide rule.

Three hundred years of dominance by the slide rule was brought to a screeching halt by the launch of the first pocket calculator in 1972.

Quadratic equations are equations with an x and an x², e.g. 3x² + 2x – 4 = 0. ‘Quadratics have become so crucial to the understanding of the world, that it is no exaggeration to say that they underpin modern science’ (p.200).

Chapter Six – Playtime

Number games. The origin of Sudoku, which is Japanese for ‘the number must appear only once’. There are some 5 billion ways for numbers to be arranged in a table of nine cells so that the sum of any row or column is the same.

There have, apparently, only been four international puzzle crazes with a mathematical slant – the tangram, the Fifteen puzzle, Rubik’s cube and Sudoku – and Bellos describes the origin and nature and solutions to all four. More than 300 million cubes have seen sold since Ernö Rubik came up with the idea in 1974. Bellos gives us the latest records set in the hyper-competitive sport of speedcubing: the current record of restoring a copletely scrambled cube to order (i.e. all the faces of one colour) is 7.08 seconds, a record held by Erik Akkersdijk, a 19-year-old Dutch student.

A visit to the annual Gathering for Gardner, honouring Martin Gardner, one of the greatest popularisers of mathematical games and puzzles who Bellos visits. The origin of the ambigram, and the computer game Tetris.

Chapter Seven – Secrets of Succession

The joy of sequences. Prime numbers.

The fundamental theorem of arithmetic – In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers.

The Goldbach conjecture – one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that, Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort.

Neil Sloane’s idea of persistence – The number of steps it takes to get to a single digit by multiplying all the digits of the preceding number to obtain a second number, then multiplying all the digits of that number to get a third number, and so on until you get down to a single digit. 88 has a persistence of three.

88 → 8 x 8 = 64 → 6 x 4 = 24 → 2 x 4 = 8

John Horton Conway’s idea of the powertrain – For any number abcd its powertrain goes to abcd, in the case of numbers with an odd number of digits the final one has no power, abcde’s powertrain is abcde.

The Recamán sequence Subtract if you can, unless a) it would result in a negative number or b) the number is already in the sequence. The result is:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11….

Gijswijt’s sequence a self-describing sequence where each term counts the maximum number of repeated blocks of numbers in the sequence immediately preceding that term.

1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, …

Perfect number A perfect number is any number that is equal to the sum of its factors. Thus 6 – its factors (the numbers which divided into it) are 1, 2 and 3. Which also add up to (are the sum of) 6. The next perfect number is 28 because its factors – 1, 2, 4, 7, 14 – add up to 28. And so on.

Amicable numbers A number is amicable if the sum of the factors of the first number equals the second number, and if the sum of the factors of the second number equals the first. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Added together these make 284. The factors of 284 are 1, 2, 4, 71 and 142. Added together they make 220!

Sociable numbers In 1918 Paul Poulet invented the term sociable numbers. ‘The members of aliquot cycles of length greater than 2 are often called sociable numbers. The smallest two such cycles have length 5 and 28’

Mersenne’s prime A prime number which can be written in the form 2n – 1 a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, … and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, …

These and every other sequence ever created by humankind are documented on The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane’s. This is an online database of integer sequences, created and maintained by Neil Sloane while a researcher at AT&T Labs.

Chapter Eight – Gold Finger

The golden section a number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

Phi The number is often symbolized using phi, after the 21st letter of the Greek alphabet. In an equation form:

a/b = (a+b)/a = 1.6180339887498948420 …

As with pi (the ratio of the circumference of a circle to its diameter), the digits go on and on, theoretically into infinity. Phi is usually rounded off to 1.618.

The Fibonnaci sequence Each number in the sequence is the sum of the two numbers that precede it. So the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn.

as the basis of seeds in flowerheads, arrangement of leaves round a stem, design of nautilus shell and much more.

Chapter Nine – Chance Is A Fine Thing

A chapter about probability and gambling.

Impossibility has a value 0, certainty a value 1, everything else is in between. Probabilities can be expressed as fractions e.g. 1/6 chance of rolling a 6 on a die, or as percentages, 16.6%, or as decimals, 0.16…

The probability is something not happening is 1 minus the probability of that thing happening.

Probability was defined and given mathematical form in 17th century. One contribution was the questions the Chevalier de Méré asked the mathematical prodigy Blaise Pascal. Pascal corresponded with his friend, Pierre de Fermat, and they worked out the bases of probability theory.

Expected value is what you can expect to get out of a bet. Bellos takes us on a tour of the usual suspects – rolling dice, tossing coins, and roulette (invented in France).

Payback percentage if you bet £10 at craps, you can expect – over time – to receive an average of about £9.86 back. In other words craps has a payback percentage of 98.6 percent. European roulette has a payback percentage of 97.3 percent. American roulette, 94.7 percent. On other words, gambling is a fancy way of giving your money away. A miserly slot machine has a payback percentage of 85%. The National Lottery has a payback percentage of 50%.

The law of large numbers The more you play a game of chance, the more likely the results will approach the statistical probability. Toss a coin three times, you might get three heads. Toss a coin a thousand times, the chances are you will get very close the statistical probability of 50% heads.

The law of very large numbers With a large enough sample, outrageous coincidences become likely.

The gambler’s fallacy The mistaken belief that, if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vice versa). In other words, that a random process becomes less random, and more predictable, the more it is repeated.

The birthday paradox The probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. (These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.) In other words you only need a group of 23 people to have an evens chance that two of them share a birthday.

The drunkard’s walk

The difficulty of attaining true randomness and the human addiction to finding meaning in anything.

The distinction between playing strategy (best strategy to win a game) and betting strategy (best strategy to maximise your winnings), not always the same.

Chapter Ten – Situation Normal

Carl Friedrich Gauss, the bell curve, normal distribution aka Gaussian distribution. Normal or Gaurrian distribution results in a bell curve. Bellos describes the invention and refinement of the bell curve (he explains that ‘the long tail’ results from a mathematician who envisioned a thin bell curve as looking like two kangaroos facing each other with their long tails heading off in opposite directions). And why

Regression to the mean – if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme. And recent devastating analyses which show how startlingly random sports achievements are, from leading baseball hitters to Simon Kuper and Stefan Szymanski’s analysis of the form of the England soccer team.

Chapter Eleven – The End of the Line

Two breakthroughs which paved the way for modern i.e. 20th century, maths: the invention of non-Euclidean geometry, specifically the concept of hyperbolic geometry. To picture this draw a triangle on a Pringle. it is recognisably a triangle but all its angles do not add up to 180°, therefore it defies, escapes, eludes all the rule of Euclidean geometry, which were designed for flat 2D surfaces.

Bellos introduces us to Daina Taimina, a maths prof at Cornell University, who invented a way of crocheting hyperbolic surfaces. The result looks curly, like curly kale or the surface of coral.

Anyway, the breakaway from flat 2-D Euclidean space led to theories about curved geometry, either convex like a sphere, or hyperbolic like the pringle. It was this notion of curved space, which paved the way for Einstein’s breakthrough ideas in the early 20th century.

The second big breakthrough was Georg Cantor’s discovery that you can have many different types of infinity. Until Cantor the mathematical tradition from the ancient Greeks to Galileo and Newton had fought shy of infinity which threatened to disrupt so many formulae.

Cantor’s breakthrough was to stop thinking about numbers, and instead think of sets. This is demonstrated through the paradoxes of Hilbert’s Hotel. You need to buckle your safety belt to understand it.


This is easily the best book about maths I’ve ever read. It gives you a panoramic history of the subject which starts with innumerate cavemen and takes us to the edge of Einstein’s great discoveries. But Bellos adds to it all kinds of levels and abilities.

He is engaging and candid and funny. He is fantastically authoritative, taking us gently into forests of daunting mathematical theory without placing a foot wrong. He’s a great explainer. He knows a good story when he sees one, and how to tell it engagingly. And in every chapter there is a ‘human angle’ as he describes his own personal meetings and interviews with many of the (living) key players in the world of contemporary maths, games and puzzles.

Like the Ian Stewart book but on a vastly bigger scale, Bellos makes you feel what it is like to be a mathematician, not just interested in nature’s patterns (the basis of Stewart’s book, Nature’s Numbers) but in the beauty of mathematical theories and discoveries for their own sakes. (This comes over very strongly in chapter seven with its description of some of the weirdest and wackiest number sequences dreamed up by the human mind.) I’ve often read scientists describing the beauty of mathematical theories, but Bellos’s book really helps you develop a feel for this kind of beauty.

For me, I think three broad conclusions emerged:

1. Most mathematicians are in it for the fun. Setting yourself, and solving, mathematical puzzles is obviously extremely rewarding. Maths includes the vast territory of puzzles and games, such as the Sudoku and so on he describes in chapter six. Obviously it has all sorts of real-world application in physics, engineering and so on, but Bellos’s book really brings over that a true understanding of maths begins in puzzles, games and patterns, and often remains there for a lifetime. Like everything else maths is no highly professionalised the property of tenured professors in universities; and yet even to this day – as throughout its history – contributions can be made by enthusiastic amateurs.

2. As he points out repeatedly, many insights which started out as the hobby horses of obsessives, or arcane breakthroughs on the borders of our understanding, and which have been airily dismissed by the professionals, often end up being useful, having applications no-one dreamed of. Either they help unravel aspects of the physical universe undreamed of when they were discovered, or have been useful to human artificers. Thus the development of random number sequences seemed utterly pointless in the 19th century, but now underlies much internet security.

On a profounder note, Bellos expresses the eerie, mystical sense many mathematicians have that it seems so strange, so pregnant with meaning, that so many of these arcane numbers end up explaining aspects of the world their inventors knew nothing of. Ian Stewart has an admirably pragmatic explanation for this: he speculates that nature uses everything it can find in order to build efficient life forms. Or, to be less teleological, over the past 3 and a half billion years, every combination of useful patterns has been tried out. Given this length of time, and the incalculable variety of life forms which have evolved on this planet, it would be strange if every number system conceivable by one of those life forms – humankind – had not been tried out at one time or another.

3. My third conclusion is that, despite John Allen Paulos’s and Bellos’s insistence, I do not live in a world ever-more bombarded by maths. I don’t gamble on anything, and I don’t follow sports – the two biggest popular areas where maths is important – and the third is the twin areas of surveys and opinion polls (55% of Americans believe in alien abductions etc etc) and the daily blizzard of reports (for example, I see in today’s paper that the ‘Number of primary school children at referral units soars’).

I register their existence but they don’t impact on me for the simple reason that I don’t believe any of them. In 1992 every opinion poll said John Major would lose the general election, but he won with a thumping majority. Since then I haven’t believed any poll about anything. For example almost all the opinion polls predicted a win for Remain in the Brexit vote. Why does any sane person believe opinion polls?

And ‘new and shocking’ reports come out at the rate of a dozen a day and, on closer examination, lots of them turn out to be recycled information, or much much more mundane releases of data sets from which journalists are paid to draw the most shocking and extreme conclusions. Some may be of fleeting interest but once you really grasp that the people reporting them to you are paid to exaggerate and horrify, you soon learn to ignore them.

If you reject or ignore these areas – sport, gambling and the news (made up of rehashed opinion polls, surveys and reports) – then unless you’re in a profession which actively requires the sophisticated manipulation of figures, I’d speculate that most of the rest of us barely come into contact with numbers from one day to the next.

I think that’s the answer to Paulos and Bellos when they are in their ‘why aren’t more people mathematically numerate?’ mode. It’s because maths is difficult, and counter-intuitive, and hard to understand and follow, it is a lot of work, it does make your head ache. Even trying to solve a simple binomial equation hurt my brain.

But I think the biggest reason that ‘we’ are so innumerate is simply that – beautiful, elegant, satisfying and thought-provoking though maths may be to the professionals – maths is more or less irrelevant to most of our day to day lives, most of the time.

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