A Monarchy Transformed: Britain 1603-1714 by Mark Kishlansky (1996)

Mark Kishlansky (1948 – 2015) was an American historian of seventeenth-century British politics. He was the Frank Baird, Jr. Professor of History at Harvard University, editor of the Journal of British Studies from 1984 to 1991, and editor-in-chief of History Compass from 2003 to 2009.

Kishlansky wrote half a dozen or so books and lots of articles about Stuart Britain and so was invited to write Volume Six of the Penguin History of England covering that period, under the general editorship of historian David Cannadine.

I think of the history of Britain in the 17th century as consisting of four parts:

  1. The first two Stuarts (Kings James I & Charles I) 1603 – 1642
  2. The Civil Wars and Protectorate (Oliver Cromwell) 1642 – 1660
  3. The Restoration (Kings Charles II & James II) 1660 – 1688
  4. The Glorious Revolution and Whig monarchs (William & Mary, then Queen Anne) 1688 – 1714

Although obviously you can go by monarch:

  1. James I (1603-25)
  2. Charles I (1625-42)
  3. Wars of the three kingdoms (1637-53)
  4. Protectorate of Oliver Cromwell (1653-1660)
  5. Charles II (1660-1685)
  6. James II and the Glorious Revolution (1685-88)
  7. William & Mary (1688-1702)

I appreciate that this is an English perspective, and Kishlansky is the first to acknowledge his history tends to focus on England, by far the largest and most powerful of the three kingdoms of Britain. The histories of Scotland and Ireland over the same period shadowed the English timeline but – obviously – had significant events, personnel and continuities of their own. From the start Kishlansky acknowledges he doesn’t have space to give these separate histories the space they deserve.

Why is the history of seventeenth century Britain so attractive and exciting?

The seventeenth century has a good claim to being the most important, the most interesting and maybe the most exciting century in English history because of the sweeping changes that affected every level of society. In 1600 England was still a late-medieval society; in 1700 it was an early modern society and in many ways the most advanced country on earth.

Social changes

  • business the modern business world was created, with the founding of the Bank of England and Lloyds insurance, cheques, banknotes and milled coins were invented; the Stock Exchange was founded and the National Debt, a financial device which allowed the British government to raise large sums for wars and colonial settlement; excise and land taxes provided reliable sources of revenue for the government
  • empire the British Empire was defined with the growth of colonies in North America and India
  • feudal forms of government withered and medieval practices such as torture and the demonisation of witchcraft and heresy died out
  • media newspapers were invented and went from weekly to daily editions
  • new consumer products domestic consumption was transformed by the arrival of new products including tobacco, sugar, rum, gin, port, champagne, tea, coffee and Cheddar cheese
  • the scientific revolution biology, chemistry and physics trace their origins to discoveries made in the 1600s – Francis Bacon laid the intellectual foundations for the scientific method; William Harvey discovered the circulation of the blood; Robert Boyle posited the existence of chemical elements, invented perfected the air pump and created the first vacuum; Isaac Newton discovered his laws of thermodynamics, the composition of light, the laws of gravity; William Napier invented logarithms; William Oughtred invented the multiplication sign in maths; Edmund Halley identified the comet which bears his name, Robert Hooke invented the microscope, the quadrant, and the marine barometer; the Royal College of Physicians published the first pharmacopeia listing the properties of drugs; Peter Chamberlen invented the forceps; the Royal Society (for the sciences) was founded in 1660
  • sport the first cricket and gold clubs were founded; Izaak Walton codified knowledge about fishing in The Compleat Angler; Charles II inaugurated yacht racing at Cowes and Queen Anne founded Royal Ascot
  • architecture Inigo Jones, Sir Christopher Wren, Nicholas Hawksmoor and John Vanbrugh created wonderful stately homes and public buildings e.g. Jones laid out the Covent Garden piazza which remains an attraction in London to this day and Wren designed the new St Paul’s cathedral which became a symbol of London
  • philosophy the political upheavals produced two masterworks of political philosophy, the Leviathan of Thomas Hobbes and John Locke’s Two Treatises of Government, which are still studied and applied in a way most previous philosophy isn’t
  • non conformists despite repeated attempts to ban them, Puritan sects who refused to ‘conform’ to the Restoration settlement of the Church of England were grudgingly accepted and went on to become a permanent and fertile element of British society – the Quakers, Baptists and Presbyterians

Political upheaval

At the centre of the century sits the great 20-year upheaval, the civil wars or British wars or Great Rebellion or the Wars of Three Kingdoms, fought between the armies of parliament and the armies of King Charles I, with significant interventions by armies of Scotland and Ireland, which eventually led to the execution of the king, the abolition of the House of Lords and the disestablishment of the Church of England – achievements which still form a core of the radical agenda to this day. These revolutionary changes were- followed by a series of constitutional experiments under the aegis of the military dictator Oliver Cromwell, which radicalised and politicised an entire generation.

Soon after Cromwell’s death in 1658, his regime began to collapse and elements of it arranged for the Restoration of King Charles II, who returned but under a new, more constitutional monarchy, restrained by laws and conventions guaranteeing the liberties of British subjects and well aware of the mistakes which led to the overthrow of his father.

But none of this stopped his overtly Roman Catholic brother, who succeeded him as James II in 1685, making a string of mistakes which collectively alienated the Protestant grandees of the land who conspired to overthrow him and replace him with the reliably Protestant Prince William of Orange. James was forced to flee, William was invited to become King of England and to rule according to a new, clearly defined constitution or Bill of Rights, which guaranteed all kinds of liberties including of speech and assembly.

All of these upheavals meant that by 1700 England had the most advanced, liberal and open society in Europe, maybe in the world, had experimented with a wide variety of political reforms and constitutions, and developed one which seemed most practical and workable – which was to become the envy of liberals in neighbouring France, and the basis of the more thoroughly worked-out Constitution devised by the founders of the American republic in the 1780s.

Studying the 17th century combines the intellectual excitement of watching these constitutional and political developments unfold, alongside the more visceral excitement of following the dramatic twists and turns in the long civil wars – and then following the slow-burning problems which led to the second great upheaval, the overthrow of James II. There is tremendous pleasure to be had from getting to know the lead characters in both stories and understanding their motives and psychologies.

Key features of 17th century England

The first two chapters of Mark Kishlansky’s book set out the social and political situation in Britain in 1600. These include:

Britain was a comprehensively patriarchal society. The king ruled the country and his word meant life or death. Le Roy le veult – the King wishes it – was the medieval French phrase still used to ratify statutes into law. The monarch made all political, legal, administrative and religious appointments – lords, ministers, bishops, judges and magistrates owed their position to him. In every locality, knights of the shires, justices of the peace administered the king’s laws. The peerage was very finely gradated and jealously policed. Status was everything.

And this hierarchy was echoed in families which were run by the male head of the household who had complete power over his wife and children, a patriarchal household structure endorsed by the examples in the Bible. Women might have as many as 9 pregnancies, of which 6 went to term and three died in infancy, with a further three children dying in infancy.

The family was primarily a unit of production, with all family members down to small children having specified tasks in the often backbreaking toil involved in agricultural work, caring for livestock, foraging for edibles in woods and fields, producing clothes and shoes. Hard physical labour was the unavoidable lot of almost the entire population.

Marriages were a vital way of passing on land and thus wealth, as well as family names and lineages. Most marriages were arranged to achieve these ends. The top responsibility of both spouses were the rights and responsibilities of marriage i.e. a wife obeyed her husband and a husband cherished and supported his wife. It was thought that ‘love’ would grow as a result of carrying out these duties, but wasn’t a necessary component.

Geography 80% of the population in 1600 worked on the land. Britain can be divided into two geographical zones:

1. The North and West The uplands of the north-west, including Scotland and Wales, whose thin soils encouraged livestock supplemented by a thin diet of oats and barley. Settlements here were scattered and people arranged themselves by kin, in Scotland by clans. Lords owned vast estates and preserved an old-fashioned medieval idea of hospitality and patronage.

Poor harvests had a catastrophic impact. A run of bad harvests in the 1690s led to mass emigration from Scotland to America, and also to the closer ‘plantations’ in Ulster.

It was at this point that Scottish Presbyterians became the majority community in the province. Whereas in the 1660s, they made up some 20% of Ulster’s population… by 1720 they were an absolute majority in Ulster, with up to 50,000 having arrived during the period 1690-1710. (Wikipedia)

2. The south and east of Britain was more densely populated, with villages and towns instead of scattered homesteads. Agriculture was more diverse and productive. Where you have more people – in towns and cities – ties of kinship become weaker and people assess each other less by ‘family’ than by achievements, social standing and wealth.

The North prided itself on its older, more traditional values. The South prided itself on being more productive and competitive.

Population The population of England rose from 4 million in 1600 to 5 million in 1700. There were maybe 600 ‘towns’ with populations of around 1,000. Big provincial capitals like Norwich, Exeter or Bristol (with pops from 10,000 to 30,000) were exceptions.

London was unlike anywhere else in Britain, with a population of 200,000 in 1600 growing to around 600,000 by 1700. It was home to the Court, government with its Houses of Lords and Commons, all the main law courts, and the financial and mercantile hub of the nation (Royal Exchange, Royal Mint, later the Bank of England and Stock Exchange). The centre of publishing and the new science, literature, the arts and theatre. By 1700 London was the largest city in the Western world. Edinburgh, the second largest city in Britain, had a paltry 40,000 population.

Inflation Rising population led to a squeeze on food since agricultural production couldn’t keep pace. This resulted in continuous inflation with foodstuffs becoming more expensive throughout the century, which reduced living standards in the countryside and contributed to periods of near famine. On the other hand, the gentry who managed to hang onto or increase their landholdings saw an unprecedented rise in their income. The rise of this class led to the development of local and regional markets and to the marketisation of agriculture. Those who did well spent lavishly, building manors and grand houses, cutting a fine figure in their coaches, sending the sons to university or the army, educating their daughters in order to attract wealthy husbands.

Vagrancy The change in working patterns on the land, plus the rising population, led to a big increase in vagrancy, which the authorities tackled with varying degrees of savagery, including branding on the face with a V for Vagrant. Contemporary theorists blamed overpopulation for poverty, vagrancy and rising crime. One solution was to encourage the excess population to settle plantations in sparsely populated Ireland or emigrate to New England. There were moral panics about rising alcoholism, and sex outside marriage.

Puritans Leading the charge to control immoral behaviour were the Puritans, a negative word applied to a range of people who believed that the Church of England needed to be further reformed in order to reach the state of purity achieved by Calvinists on the continent. Their aims included:

  • abolition of the 26 bishops (who were appointed by the king) and their replacement by Elders elected by congregations
  • reforms of theology and practice – getting rid of images, candles, carvings etc inside churches, getting rid of elaborate ceremonies, bells and incense and other ‘Roman’ superstitions
  • reducing the number of sacraments to the only two practiced by Jesus in the New Testament
  • adult baptism replacing infant baptism

Banning Closely connected was the impulse to crack down on all ungodly behaviour e.g. alcohol (close pubs), immorality (close theatres), licentiousness (ban most books except the Bible), lewd behaviour (force women to wear modest outfits, keep their eyes on the ground), ban festivals, ban Christmas, and so on.

Trans-shipping The key driver of Britain’s economic wealth was shipping and more precisely trans-shipping – where goods were brought in from one source before being transhipped on elsewhere. The size of Britain’s merchant fleet more than tripled and the sized of the cargo ships increased tenfold. London’s wealth was based on the trans-shipping trade.

The end of consensus politics

The second of Kishlansky’s introductory chapters describes in detail the political and administrative system in early 17th century Britain. It is fascinating about a) the complexity of the system b) its highly personal orientation about the person the monarch. It’s far too complicated to summarise here but a few key themes emerge:

Consensus Decisions at every level were reached by consensus. To give an example, when a new Parliament was called by the king, the justices of the peace in a county met at a session where, usually, two candidates put themselves forward and the assembled JPs discussed and chose one. Only very rarely were they forced back on the expedient of consulting local householders i.e. actually having a vote on the matter.

Kishlansky explains how this principle of consensus applied in lots of other areas of administration and politics, for example in discussions in Parliament about acts proposed by the king and which needed to be agreed by both Commons and Lords.

He then goes on to launch what is – for me at any rate – a new and massive idea: that the entire 17th century can be seen as the slow and very painful progression from a political model of consensus to an adversarial model.

The entire sequence of civil war, dictatorship, restoration and overthrow can be interpreted as a series of attempts to reach a consensus by excluding your opponents. King Charles prorogued Parliament to get his way, then tried to arrest its leading members. Cromwell, notoriously, was forced to continually remodel and eventually handpick a Parliament which would agree to do his bidding. After the Restoration Charles II tried to exclude both Catholics and non-conforming Protestants from the body politic, imposing an oath of allegiance in order to preserve the model of consensus sought by his grandfather and father.

the point is that all these attempts to purify the body politic in order to achieve consensus failed.

The advent of William of Orange and the Bill of Rights in 1689 can be seen as not so much defining liberties and freedoms but as finally accepting the new reality, that political consensus was no longer possible and only a well-managed adversarial system could work in a modern mixed society.

Religion What made consensus increasingly impossible? Religion. The reformation of Roman Catholicism which began in 1517, and continued throughout the 16th century meant that, by the 1620s, British society was no longer one culturally and religiously unified community, but included irreducible minorities of Catholics and new-style Calvinist Puritans. Both sides in what became the civil wars tried to preserve the old-fashioned consensus by excluding what they saw as disruptive elements who prevented consensus agreements being reached i.e. the Royalists tried to exclude the Parliamentarians, the Parliamentarians tried to exclude the Royalists, both of them tried to exclude Catholics, the Puritans once in power tried to exclude the Anglicans and so on.

But the consensus model was based on the notion that, deep down, all participants shared the same religious, cultural and social values. Once they had ceased to do that the model was doomed.

Seen from this point of view the entire history of the 17th century was the slow, bloody, and very reluctant acceptance that the old model was dead and that an entirely new model was required in which political elites simply had to accept the long-term existence of sincere and loyal but completely different opinions from their own.

Political parties It is no accident that it was after the Glorious Revolution that the seeds of what became political parties first began to emerge. Under the consensus model they weren’t needed; grandees and royal ministers and so on managed affairs so that most of them agreed or acquiesced on the big decisions. Political parties only become necessary or possible once it had become widely accepted that consensus was no longer possible and that one side or another in a debate over policy would simply lose and would have to put up with losing.

So Kishlansky’s long and fascinating introduction leads up to this insight – that the succession of rebellions and civil wars across the three kingdoms, the instability of the Restoration and then the overthrow of James II were all necessary to utterly and finally discredit the old late-medieval notion of political decision-making by consensus, and to usher in the new world of political decision-making by votes, by parties, by lobbying, by organising, by arguing and taking your arguments to a broader political nation i.e. the electorate.

In large part the English Revolution resulted from the inability of the consensual political system to accommodate principled dissension. (p.63)

At a deep level, the adoption of democracy means the abandonment of attempts to repress a society into agreement. On this view, the core meaning of democracy isn’t the paraphernalia about voting, that’s secondary. In its essence democracy means accepting other people’s right to disagree, sincerely and deeply, with what you hold to be profoundly true. Crafting a system which allows people to think differently and speak differently and live differently, without fear or intimidation.

Related links

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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Walk: Albury Park

7 May 2012

Just as I stepped off the train at Clandon it started to rain. I thought I’d figured a neat short cut to Albury without quite realising it involved cycling over the North Downs. In the rain. With the wind in my face. Still it was downhill on the other side to Silent Pool, where I locked the bike and strolled through the Victorian village of Albury, all decorated brick, mock Tudor chimneys and – if you looked closely enough – wild flowers in the rain.

Forget-me-nots in Albury

Forget-me-nots in Albury

Up the hillside to the Victorian church of St Peter and St Paul. Strange how ugly Victorian churches can be. A pile of red bricks surrounded by dismal cracked flagstones, it felt like a factory or a workhouse. Reminded me of the horrible brick church in the village where I grew up, Chavey Down. And the vast empty barn of a church round the corner from me, St Thomas’s, Streatham Hill. But in the rainy churchyard there were primroses and cowslips.

Primroses in the graveyard of St Peter's church, Albury

Primroses in the graveyard of St Peter’s church Albury

Up a deep muddy country lane in the rain to Albury Warren, conifer woods at the top, then through a gate into the 150 acre grounds of Albury Park, still dominated by  its Victorian mansion, the hillside landscaped with rhododendrons, and more flowers: I saw goose grass, dog’s mercury, white dead nettle, archangel, scads of dandelions but not many bluebells.

Wood cranesbill in Albury Warren

Wood cranesbill in Albury Warren

Finally, I escaped the rain in the historic Saxon church of St Peter and St Paul. This used to be the heart of the village till the early Victorian landowner turfed the villagers out, rebuilding their village a mile to the West. That explains why modern Albury is so Victorian in feel, and explains the horrible ‘new’ Albury church he built for them. He let the original medieval church slowly decay, till it was saved and restored in the 1920s and is now open to visitors, bare empty inside, except for a rare medieval wall painting – of St Christopher – and the florid family chapel designed by Augustus Pugin.

William Oughtred was rector here for 50 years in the 17th century. Who he? The leading mathemetician of his day who invented the slide rule in 1622, introduced the ‘x’ symbol for multiplication, and was tutor to Sir Christopher Wren. All that and a sermon every Sunday!

Moreover, Robert Malthus, the man who invented the gloomy Malthusian economics which dominated Victorian England, wrote his famous book here, ‘An Essay on the Principle of Population’. It’s well worth reading in order to grasp the impact it had over the entire succeeding century. It was one spur for the drafting of the Poor Laws which led to the Victorian Workhouses which Dickens so railed against, and which Albury church so balefully reminded me of.

Malthus’s impact was felt not only here but in Britain’s Imperial colonies. In his wonderful book on Kipling, Charles Allen points out that it was the insistence of the Viceroy to India, Lord Lytton, appointed by Disraeli, that doctrinaire free market and Malthusian principles were followed during the famines of the later 1870s – directly causing the deaths of hundreds of thousands of Indians from starvation – that led to the founding of the Indian National Congress and the beginnings of the struggle for independence. Malthus hovered over all Victorian thought like the threat of nuclear annihilation dominated the later 20th century…

England has such depth, such resonance.  All this history and significance packed into a little stone building by a tiny gurgling stream (the ‘river’ Tilling). And the pretty flowers, blowing all around in the steady English rain…

Greater stitchwort, Albury Park

Greater stitchwort, Albury Park

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