Mathematics how it shaped our world

‘There were no programmes, no calculating machines … we relied upon our slide rules and arithmetic in the margins … Lives were at stake and we couldn’t afford to let anything go through wrong.’

Letitia Chitty recalling Admiralty Air Department work in the First World War1

The aviation industry was a popular employer for Britain’s top mathematicians during the First World War, trained in the mathematics of flight at places such as Imperial College London and Cambridge University, which was described by the mathematics historian June Barrow-Green as ‘the cradle of the nation’s mathematical research’.2 As in the aircraft industry in general, a number of those working in aeronautics were women, such as Letitia Chitty, whose comment on the importance of their mathematical work opens this introduction.

Chitty read mathematics at Newnham College, Cambridge, before working for the Admiralty Air Department. After the war ended, she moved briefly to the Bristol Aeroplane Company before

beginning a career at Imperial College as a mathematician and civil engineer. Her name is little remembered today outside the narrow confines of aeronautical history and her work might appear at odds with the old received histories of mathematics, which tended to focus on theorems and great men. Yet, as she commented, lives depended on her work and that of her colleagues, and anybody travelling by aeroplane today can be grateful to them for their mathematical skill.

The case of aeronautics helps set out the approach of this book, and the Mathematics Gallery on which it is based. Rather than being a book about mathematical theorems or ideas, it looks at mathematical practice, the work of a very broad range of people that we might call mathematical in one way or another – or that the people themselves might have considered mathematical, because the scope encompassed by the term has changed over time. It is a series of case studies about the role of mathematical practice in the development of the modern world over the last 400 years. We will return to the case study of aviation presently.

It is in the choice of these case studies that the book might claim some novelty. As a publication to accompany a major museum display, it looks at that mathematical practice through particular objects held in the Science Museum’s collections, augmented by a few choice additions. It is a history in objects.

Which objects, though? There are plenty of museum catalogues Admiralty Air Department, 1918 Engineers, mathematicians and designers working at the Admiralty’s Air Department at the end of the First World War. Letitia Chitty is third from the left in the middle row.

10

M a t h e ma t i c s

Architectural drawing class, St Petersburg, early twentieth century Since its founding, architecture has been a mathematical practice, and architectural spaces embody mathematical ideas.

of mathematical instruments and models, which could be defined as the conventional tools of mathematicians and the artefacts they sometimes make to express ideas. But we can take the idea of mathematical objects much further. When people practise anything, they are influenced by the material world and they leave imprints on it in turn. It is that two-way process – of the real world influencing the work of mathematicians and also being shaped by it – that is the focus of both the Gallery and this book. This is a much wider approach to a ‘material culture’ of mathematics than tools and models, though they do feature here.

So, in this book, ‘mathematics’ means a broad range of mathematical practice, and ‘mathematician’ is not just the current, rather narrow definition of the term (which tends to refer to those working in universities), but a much wider understanding that takes in more people than usual, from salespeople to sailors, aircraft engineers to insurance clerks, architects to bankers, gamblers to garden designers, neurologists to cryptanalysts, coders to traders.

All may be thought of as mathematicians in a historical sense and all have interesting stories to tell.

The historian Stephen Johnston has looked at the situation that existed over 400 years ago as follows: The Renaissance conception of mathematics was of a very broad church indeed. Mathematics then incorporated not only elements that we would now recognise as mathematics but a whole host of other activities and arts that today are seen as belonging to science and technology: astronomy, navigation, magnetism, surveying, architecture, shipbuilding, artillery, the list goes on and on.3

Detail from a slide rule designed

by William Froude, 1877 This circular slide rule was designed by hydrodynamicist Froude to help understand water-flow around ships.

The underlying equations remain one of the biggest challenges in pure mathematics.

Object No: 1941-8/1

introduction

11

‘There were no programmes, no calculating machines … we relied upon our slide rules and arithmetic in the margins … Lives were at stake and we couldn’t afford to let anything go through wrong.’

Letitia Chitty recalling Admiralty Air Department work in the First World War1

The aviation industry was a popular employer for Britain’s top mathematicians during the First World War, trained in the mathematics of flight at places such as Imperial College London and Cambridge University, which was described by the mathematics historian June Barrow-Green as ‘the cradle of the nation’s mathematical research’.2 As in the aircraft industry in general, a number of those working in aeronautics were women, such as Letitia Chitty, whose comment on the importance of their mathematical work opens this introduction.

Chitty read mathematics at Newnham College, Cambridge, before working for the Admiralty Air Department. After the war ended, she moved briefly to the Bristol Aeroplane Company before

beginning a career at Imperial College as a mathematician and civil engineer. Her name is little remembered today outside the narrow confines of aeronautical history and her work might appear at odds with the old received histories of mathematics, which tended to focus on theorems and great men. Yet, as she commented, lives depended on her work and that of her colleagues, and anybody travelling by aeroplane today can be grateful to them for their mathematical skill.

The case of aeronautics helps set out the approach of this book, and the Mathematics Gallery on which it is based. Rather than being a book about mathematical theorems or ideas, it looks at mathematical practice, the work of a very broad range of people that we might call mathematical in one way or another – or that the people themselves might have considered mathematical, because the scope encompassed by the term has changed over time. It is a series of case studies about the role of mathematical practice in the development of the modern world over the last 400 years. We will return to the case study of aviation presently.

It is in the choice of these case studies that the book might claim some novelty. As a publication to accompany a major museum display, it looks at that mathematical practice through particular objects held in the Science Museum’s collections, augmented by a few choice additions. It is a history in objects.

Which objects, though? There are plenty of museum catalogues Admiralty Air Department, 1918 Engineers, mathematicians and designers working at the Admiralty’s Air Department at the end of the First World War. Letitia Chitty is third from the left in the middle row.

10

M a t h e ma t i c s

Architectural drawing class, St Petersburg, early twentieth century Since its founding, architecture has been a mathematical practice, and architectural spaces embody mathematical ideas.

of mathematical instruments and models, which could be defined as the conventional tools of mathematicians and the artefacts they sometimes make to express ideas. But we can take the idea of mathematical objects much further. When people practise anything, they are influenced by the material world and they leave imprints on it in turn. It is that two-way process – of the real world influencing the work of mathematicians and also being shaped by it – that is the focus of both the Gallery and this book. This is a much wider approach to a ‘material culture’ of mathematics than tools and models, though they do feature here.

So, in this book, ‘mathematics’ means a broad range of mathematical practice, and ‘mathematician’ is not just the current, rather narrow definition of the term (which tends to refer to those working in universities), but a much wider understanding that takes in more people than usual, from salespeople to sailors, aircraft engineers to insurance clerks, architects to bankers, gamblers to garden designers, neurologists to cryptanalysts, coders to traders.

All may be thought of as mathematicians in a historical sense and all have interesting stories to tell.

The historian Stephen Johnston has looked at the situation that existed over 400 years ago as follows: The Renaissance conception of mathematics was of a very broad church indeed. Mathematics then incorporated not only elements that we would now recognise as mathematics but a whole host of other activities and arts that today are seen as belonging to science and technology: astronomy, navigation, magnetism, surveying, architecture, shipbuilding, artillery, the list goes on and on.3

Detail from a slide rule designed

by William Froude, 1877 This circular slide rule was designed by hydrodynamicist Froude to help understand water-flow around ships.

The underlying equations remain one of the biggest challenges in pure mathematics.

Object No: 1941-8/1

introduction

11

The world can seem a much smaller place when we trade with other countries and travel around the globe. Empires have been built and fortunes made and lost on the back of overseas trade. In a world of maritime travel, some of the greatest challenges for politicians, scientists, inventers and entrepreneurs have involved the design of more efficient ships and new techniques to navigate safely around the globe. Given that mathematicians study the properties of distance and quantity, there is little wonder that mathematics, for centuries, has underpinned our ability to trade and travel.

At the heart of trade is measurement. We take a quantity of goods and we give a quantity of money (or other goods). But across the vastness of the globe, with local measurement systems built up over hundreds of years, how do we know how much of anything we are really buying? Without a means of comparing weight, volume and length measurements with those of different nations or regions, successful trade was severely hampered. A standard measurement system – or at least one where comparisons between systems were known with mathematical precision – was vital in the drive to build trading empires.

But a common language of measurement was only one difficulty in the building of trade networks. Another of the greatest challenges of the seventeenth and eighteenth centuries was known as the longitude problem, in which maritime navigation was unreliable and unsafe as navigators had no accurate way of locating their east–west position, or longitude. Mathematics helped provide an answer with the ‘lunar-distance’ method of finding longitude.

Alongside advances in navigation, trade continued to expand rapidly with ever-larger ships carrying ever-greater quantities of goods around the world. The ascendency of the steam-powered ship in the nineteenth century transformed maritime trade but introduced a major new cost: fuel. To keep profits up, attention turned to the design of ships: to reduce friction and increase fuel-efficiency. Mathematics took its place alongside engineering experimentation in the design of new shapes for ships.

Detail from a model of an East India Company ship, c.1809 Object No: 1929-406

18

M a t h e ma t i c s

Yet commercial success has not just depended on the physical trade and movement of goods. It has always also relied on the ability to transmit commercial information in secret over long distances. Mathematics has long been put to use making codes and ciphers, as well as in the breaking of them. From simple encoding systems to elaborate electro-mechanical cipher machines and today’s computer algorithms, cryptography looks set to keep mathematicians in employment for as long as we have a need to keep secrets.

CH A P TE R 1

tr ad e an d tr avel

19

The world can seem a much smaller place when we trade with other countries and travel around the globe. Empires have been built and fortunes made and lost on the back of overseas trade. In a world of maritime travel, some of the greatest challenges for politicians, scientists, inventers and entrepreneurs have involved the design of more efficient ships and new techniques to navigate safely around the globe. Given that mathematicians study the properties of distance and quantity, there is little wonder that mathematics, for centuries, has underpinned our ability to trade and travel.

At the heart of trade is measurement. We take a quantity of goods and we give a quantity of money (or other goods). But across the vastness of the globe, with local measurement systems built up over hundreds of years, how do we know how much of anything we are really buying? Without a means of comparing weight, volume and length measurements with those of different nations or regions, successful trade was severely hampered. A standard measurement system – or at least one where comparisons between systems were known with mathematical precision – was vital in the drive to build trading empires.

But a common language of measurement was only one difficulty in the building of trade networks. Another of the greatest challenges of the seventeenth and eighteenth centuries was known as the longitude problem, in which maritime navigation was unreliable and unsafe as navigators had no accurate way of locating their east–west position, or longitude. Mathematics helped provide an answer with the ‘lunar-distance’ method of finding longitude.

Alongside advances in navigation, trade continued to expand rapidly with ever-larger ships carrying ever-greater quantities of goods around the world. The ascendency of the steam-powered ship in the nineteenth century transformed maritime trade but introduced a major new cost: fuel. To keep profits up, attention turned to the design of ships: to reduce friction and increase fuel-efficiency. Mathematics took its place alongside engineering experimentation in the design of new shapes for ships.

Detail from a model of an East India Company ship, c.1809 Object No: 1929-406

18

M a t h e ma t i c s

Yet commercial success has not just depended on the physical trade and movement of goods. It has always also relied on the ability to transmit commercial information in secret over long distances. Mathematics has long been put to use making codes and ciphers, as well as in the breaking of them. From simple encoding systems to elaborate electro-mechanical cipher machines and today’s computer algorithms, cryptography looks set to keep mathematicians in employment for as long as we have a need to keep secrets.

CH A P TE R 1

tr ad e an d tr avel

19

the shape of ships ‘The resistance to motion of a good ship-form has been improved by about 10 per cent. The effect comes back to shipowners as a direct saving in fuel consumption amounting, on a very conservative estimate, to over £1,000,000 a year.’

The Times on the opening of a ship test-tank at the National Physical Laboratory, 1932 15

The gigantic oil tanker Globtik Tokyo was being readied to float out of its cavernous construction dock in Japan’s Ishikawajima-Harima shipyard in Kure, near Hiroshima. Watching it was a huge crowd made up of the men and women who had constructed it over the previous six months, along with their families. It was October 1972 and Globtik Tokyo was the largest ship in the world. Designed to carry over 500 million litres of crude oil on each journey, it was riding a wave of commercial maritime expansion fuelled by cheap oil and increasing consumer spending.

The statistics of today’s maritime trade make for sobering reading. In 2014, almost 10 billion tons of cargo were carried by ships each year: 80 per cent of all world trade by volume and over 70 per cent by value. There were some 50,000 cargo ships afloat, transporting vast quantities of crude oil, petroleum, gas, coal, iron and aluminium ore, metals, minerals, grain and phosphate rock used for fertiliser, animal feeds and the chemical industry, as well as other bulk materials.

right Globtik Tokyo under construction, 1972 A few days before Globtik Tokyo was floated out of its construction dock in the Japanese Ishikawajima-Harima shipyard, the families of workers were invited to view the huge ship.

In addition, a fleet of specialist ships carried between them some 160 million containers, the largest ship transporting the equivalent of over 19,000 standard-sized containers on each journey.16 Since containerisation took off in the 1950s it has become cheaper to manufacture goods on the other side of the world and ship them to where they are needed than it is to make them close to the place of their end use.

opposite Model of oil tanker Globtik Tokyo, 1973 When it was completed in 1973, the Globtik Tokyo – at over one-third of a kilometre (364 yards) long – was the largest ship in the world. Designed to carry crude oil, it was one of a new breed of ‘ultra-large crude carriers’ built in the early 1970s.

Object No: 2006-30

34

M a t h e ma t i c s

t rad e a n d t ra v e l

35

the shape of ships ‘The resistance to motion of a good ship-form has been improved by about 10 per cent. The effect comes back to shipowners as a direct saving in fuel consumption amounting, on a very conservative estimate, to over £1,000,000 a year.’

The Times on the opening of a ship test-tank at the National Physical Laboratory, 1932 15

The gigantic oil tanker Globtik Tokyo was being readied to float out of its cavernous construction dock in Japan’s Ishikawajima-Harima shipyard in Kure, near Hiroshima. Watching it was a huge crowd made up of the men and women who had constructed it over the previous six months, along with their families. It was October 1972 and Globtik Tokyo was the largest ship in the world. Designed to carry over 500 million litres of crude oil on each journey, it was riding a wave of commercial maritime expansion fuelled by cheap oil and increasing consumer spending.

The statistics of today’s maritime trade make for sobering reading. In 2014, almost 10 billion tons of cargo were carried by ships each year: 80 per cent of all world trade by volume and over 70 per cent by value. There were some 50,000 cargo ships afloat, transporting vast quantities of crude oil, petroleum, gas, coal, iron and aluminium ore, metals, minerals, grain and phosphate rock used for fertiliser, animal feeds and the chemical industry, as well as other bulk materials.

right Globtik Tokyo under construction, 1972 A few days before Globtik Tokyo was floated out of its construction dock in the Japanese Ishikawajima-Harima shipyard, the families of workers were invited to view the huge ship.

In addition, a fleet of specialist ships carried between them some 160 million containers, the largest ship transporting the equivalent of over 19,000 standard-sized containers on each journey.16 Since containerisation took off in the 1950s it has become cheaper to manufacture goods on the other side of the world and ship them to where they are needed than it is to make them close to the place of their end use.

opposite Model of oil tanker Globtik Tokyo, 1973 When it was completed in 1973, the Globtik Tokyo – at over one-third of a kilometre (364 yards) long – was the largest ship in the world. Designed to carry crude oil, it was one of a new breed of ‘ultra-large crude carriers’ built in the early 1970s.

Object No: 2006-30

34

M a t h e ma t i c s

t rad e a n d t ra v e l

35

Mortality ‘The contemplation of the mortality of mankind has, besides the moral, its physical and political uses.’

Edmond Halley astronomer, 16932

When are you going to die? Mathematics does not get much closer to life and death than when it is calculating our life expectancy, and for many of us this will be difficult to think about.

Most insurance products provide financial assistance in the event of something unlikely happening: a fire, a traffic collision and so on.

But one of the oldest products of all is based on an event that is certain to happen, we just do not know exactly when: our death.

Of course, each individual life follows a path that cannot be predicted. We take our chances and sometimes they pay off. But by adding together all those individual lives, we can spot certain trends that can be analysed using the mathematical tools of probability and statistics. This is the life expectancy of populations, or of sections of the population. When are we likely to die?

Many factors affect how long we are likely to live. The circumstances of our birth and upbringing, our income, our gender, our education, our diet, whether or not we smoke and drink, our occupation and our age – all these factors and many more tend to affect our life expectancy. This matters, not just for ourselves but for the society we live in. The statistical study of populations is known as ‘demography’ and affects everything from pensions to housing, and from public health and medical care to schools and employment. Many of us buy products such as life assurance or health insurance, which must be priced according to the risk we present. We need to make decisions about retirement and many other life choices. This is all based on a sophisticated understanding of the mathematics of mortality.

It was the astronomer, Edmond Halley, who first formally described the mathematics of life expectancy. The year was 1693 and Halley wanted to solve two problems. The first was how to set

96

M a t h e ma t i c s

opposite ‘An Estimate of the Degrees of the Mortality of Mankind ...’,

by Edmond Halley, 1693 Halley’s paper, published in the Royal Society’s Philosophical Transactions journal, was the first to put the pricing of life annuities and insurance products onto a rigorous mathematical footing.

the right price for an insurance product known as a ‘life annuity’, which paid an annual allowance every year the insured person was alive. Charge too much and people would choose not to take out the product. Charge too little and the scheme would lose money as people received more than they paid in. The second problem was how to work out the correct price for a life assurance premium, a product that paid out on the death of the policyholder, enabling people to provide for their families. It was vital in both cases to get the price right.

It might seem odd that somebody who spent his life gazing into the heavens was a pioneer in the very earthly study of insurance, but in the seventeenth century people we now call

Trial model of Charles Babbage’s difference engine, 1871–1900 This is one of about six demonstration models of part of Babbage’s proposed difference engine, manufactured after his death in 1871 by his son, Henry.

Object No: 1967-70

‘scientists’ applied their scientific and mathematical skills to wide ranges of problems in an attempt to transform society from superstition and tradition to reason and evidence. Halley was just one of many who helped build a new mathematics of probability.

He, more than most, had reason to ponder the probabilities of life expectancy, as his father had been brutally murdered in 1684.

What price life?

In his 1693 paper on mortality, published by the Royal Society, Halley built on the work of early demography specialists including John Graunt and William Petty, who used ‘bills of mortality’ (weekly published figures for deaths in a particular location) to look for patterns. Earlier studies had looked at the bills

CH A P TE R 4

l i f e a n d d e at h

97

Mortality ‘The contemplation of the mortality of mankind has, besides the moral, its physical and political uses.’

Edmond Halley astronomer, 16932

When are you going to die? Mathematics does not get much closer to life and death than when it is calculating our life expectancy, and for many of us this will be difficult to think about.

Most insurance products provide financial assistance in the event of something unlikely happening: a fire, a traffic collision and so on.

But one of the oldest products of all is based on an event that is certain to happen, we just do not know exactly when: our death.

Of course, each individual life follows a path that cannot be predicted. We take our chances and sometimes they pay off. But by adding together all those individual lives, we can spot certain trends that can be analysed using the mathematical tools of probability and statistics. This is the life expectancy of populations, or of sections of the population. When are we likely to die?

Many factors affect how long we are likely to live. The circumstances of our birth and upbringing, our income, our gender, our education, our diet, whether or not we smoke and drink, our occupation and our age – all these factors and many more tend to affect our life expectancy. This matters, not just for ourselves but for the society we live in. The statistical study of populations is known as ‘demography’ and affects everything from pensions to housing, and from public health and medical care to schools and employment. Many of us buy products such as life assurance or health insurance, which must be priced according to the risk we present. We need to make decisions about retirement and many other life choices. This is all based on a sophisticated understanding of the mathematics of mortality.

It was the astronomer, Edmond Halley, who first formally described the mathematics of life expectancy. The year was 1693 and Halley wanted to solve two problems. The first was how to set

96

M a t h e ma t i c s

opposite ‘An Estimate of the Degrees of the Mortality of Mankind ...’,

by Edmond Halley, 1693 Halley’s paper, published in the Royal Society’s Philosophical Transactions journal, was the first to put the pricing of life annuities and insurance products onto a rigorous mathematical footing.

the right price for an insurance product known as a ‘life annuity’, which paid an annual allowance every year the insured person was alive. Charge too much and people would choose not to take out the product. Charge too little and the scheme would lose money as people received more than they paid in. The second problem was how to work out the correct price for a life assurance premium, a product that paid out on the death of the policyholder, enabling people to provide for their families. It was vital in both cases to get the price right.

It might seem odd that somebody who spent his life gazing into the heavens was a pioneer in the very earthly study of insurance, but in the seventeenth century people we now call

Trial model of Charles Babbage’s difference engine, 1871–1900 This is one of about six demonstration models of part of Babbage’s proposed difference engine, manufactured after his death in 1871 by his son, Henry.

Object No: 1967-70

‘scientists’ applied their scientific and mathematical skills to wide ranges of problems in an attempt to transform society from superstition and tradition to reason and evidence. Halley was just one of many who helped build a new mathematics of probability.

He, more than most, had reason to ponder the probabilities of life expectancy, as his father had been brutally murdered in 1684.

What price life?

In his 1693 paper on mortality, published by the Royal Society, Halley built on the work of early demography specialists including John Graunt and William Petty, who used ‘bills of mortality’ (weekly published figures for deaths in a particular location) to look for patterns. Earlier studies had looked at the bills

CH A P TE R 4

l i f e a n d d e at h

97

Measuring people ‘No sane and experienced person can doubt the enormous difference between the natural gifts of different men, whether in moral power, in taste, in intellect, or in physical endowments.’

Francis Galton statistician and eugenicist, 189021

Anthropometry is the measurement and classification of human beings. People have long thought that measuring people can help us understand the way they think, behave and act in society. Clearly, this would be a powerful tool for those seeking to govern and control society. So, anthropometry is a mathematical activity but also a political one.

One early practice on the road to anthropometry was termed ‘phrenology’, meaning ‘the science of the mind’, invented in the 1790s by an Austrian doctor, Franz Joseph Gall. Gall worked with a medical student, Johann Caspar Spurzheim, to develop the idea that people’s personalities might be linked to the physical structure of the brain, and the strength of each of dozens of characteristics – such as compassion, arrogance, valour and ambition – would be reflected by the size of that part of the brain, which was divided into ‘organs’.

If true, this would mean that the shape of the skull – which would follow that of the brain underneath – could be read by phrenologists to assess a person’s character. For instance, a bump over the part of the brain believed to cause acquisitiveness might mean the subject was likely to be a thief. Critics sometimes dismissively called it ‘bumpology’.

After a brief period in the early nineteenth century when it was popular with doctors, phrenology went on mainly to be used by members of the public to try to make sense of their lives.

Lecturers at town and mechanics’ institutes would use plaster casts of actual heads – often as sets of ‘felons’ and ‘worthies’ – to demonstrate the principles of phrenology at their talks. There were more than 200 phrenological lecturers active in the first half of the nineteenth century. Other phrenologists used real human skulls,

108

M a t h e ma t i c s

Human skull with phrenological markings, nineteenth century Franz Joseph Gall, the inventor of phrenology, amassed a collection of some 300 human skulls, many from convicted criminals. This led to a widespread trade in phrenology skulls as well as plaster and ceramic heads.

opposite Poster explaining the concepts of phrenology, c.1845 Developed by the Austrian physician Franz Joseph Gall in the 1790s, phrenology used the size and shape of people’s heads to try to understand their mental characteristics.

Object No: 1999-398

t rad e a n d t ra v e l

109

Measuring people ‘No sane and experienced person can doubt the enormous difference between the natural gifts of different men, whether in moral power, in taste, in intellect, or in physical endowments.’

Francis Galton statistician and eugenicist, 189021

Anthropometry is the measurement and classification of human beings. People have long thought that measuring people can help us understand the way they think, behave and act in society. Clearly, this would be a powerful tool for those seeking to govern and control society. So, anthropometry is a mathematical activity but also a political one.

One early practice on the road to anthropometry was termed ‘phrenology’, meaning ‘the science of the mind’, invented in the 1790s by an Austrian doctor, Franz Joseph Gall. Gall worked with a medical student, Johann Caspar Spurzheim, to develop the idea that people’s personalities might be linked to the physical structure of the brain, and the strength of each of dozens of characteristics – such as compassion, arrogance, valour and ambition – would be reflected by the size of that part of the brain, which was divided into ‘organs’.

If true, this would mean that the shape of the skull – which would follow that of the brain underneath – could be read by phrenologists to assess a person’s character. For instance, a bump over the part of the brain believed to cause acquisitiveness might mean the subject was likely to be a thief. Critics sometimes dismissively called it ‘bumpology’.

After a brief period in the early nineteenth century when it was popular with doctors, phrenology went on mainly to be used by members of the public to try to make sense of their lives.

Lecturers at town and mechanics’ institutes would use plaster casts of actual heads – often as sets of ‘felons’ and ‘worthies’ – to demonstrate the principles of phrenology at their talks. There were more than 200 phrenological lecturers active in the first half of the nineteenth century. Other phrenologists used real human skulls,

108

M a t h e ma t i c s

Human skull with phrenological markings, nineteenth century Franz Joseph Gall, the inventor of phrenology, amassed a collection of some 300 human skulls, many from convicted criminals. This led to a widespread trade in phrenology skulls as well as plaster and ceramic heads.

opposite Poster explaining the concepts of phrenology, c.1845 Developed by the Austrian physician Franz Joseph Gall in the 1790s, phrenology used the size and shape of people’s heads to try to understand their mental characteristics.

Object No: 1999-398

t rad e a n d t ra v e l

109

How buildings stay up ‘Engineers … have applied logical reasoning and mathematical methods in questions which have previously been settled by empirical formulae and rule of thumb.’

The Times engineering correspondent, 190821

In 1995, Channel 4 television aired a landmark series about structural engineering entitled The Limit. It explored the work of people pushing engineering to the boundaries of scientific knowledge, making longer bridges, deeper tunnels, larger aircraft, taller buildings, faster ships and space technology that could go farther than anything hitherto. And it raised an important point about how we see engineers and their practice.

Its producer, Martin Mortimore, commented on the place of engineers in today’s society compared with the well-known names from the past such as Thomas Telford, George Stephenson and Isambard Kingdom Brunel. ‘Somehow they seem to have been more in touch with their projects’, he remarked, ‘holding a slide rule in one hand and scribbling equations with the other.

Today, high-powered computer programs can solve many of the equations, and even come up with the designs’. However, he concluded, ‘there are still people of vision and imagination who can push back boundaries’.22 These comments were prescient. In the widest sense they were asking why we know more about nineteenth-century engineers than about those among us today. In part, this is owing to the way the history of engineering has been written in the past.

It has made heroes out of individuals, leaving the backroom women and men unnamed and under-appreciated. But Mortimore also raised an important point about the role of mathematics in the practice and identity of engineers. Are engineers mathematicians?

And how has the answer to this question shifted over the past few hundred years?

In the distant past, it is true to say that much engineering, particularly of large-scale structures and buildings, relied more

138

M a t h e ma t i c s

Sir Marc Isambard Brunel, c.1802 This portrait is typical of early nineteenth-century depictions of engineers, with an example of their work in the background and the instruments of engineering on drawings in the foreground. Those instruments – here, dividers and a set square – could equally belong to an architect, surveyor, navigator or mathematician. Marc Isambard Brunel was the father of Isambard Kingdom Brunel.

on rules of thumb. Sometimes this involved what we might today call over-engineering, or rather building with big margins of safety because the mathematics of the structure’s forces and materials, and the behaviour of the ground on which it was built, were not known well enough. At other times, buildings needed modification after construction to strengthen parts that were too weak having been under-engineered. This practical approach based on experience was empirical rather than theoretical. Some remarkable forms could be built in this way. Look at medieval cathedrals and chapels, with their soaring vaulted roofs and flying buttresses.

Their builders had no slide rules, but the structures were incredibly daring nonetheless.

But from the seventeenth century onwards, sweeping developments in science and mathematics put our understanding

of how structures behave – the forces, stresses and strains within them – onto a formal theoretical footing. Some of the most well-known names in science worked on what we now know as structural engineering, from Galileo Galilei and Isaac Newton to Robert Hooke, with Leonhard Euler, Claude-Louis Navier and Augustin-Louis Cauchy developing sophisticated mathematical theories still used extensively today. Instrument-makers developed similarly powerful tools, such as the slide rule invented in the 1620s. Taken together, this gave engineers both a theoretical and a practical mathematical toolkit to inform their practice.

So, in Telford’s day, and that of Stephenson and Brunel – all three flourished in the first half of the nineteenth century – the answer to the question ‘are engineers mathematicians’ would have been unquestionably ‘yes’. (They would also have called

The choir and vaulted roof of St George’s Chapel, Windsor, photographed c.1890 St George’s Chapel, Windsor, is one of the world’s finest examples of medieval architecture. Its almost flat vaulted roof is breathtaking, and the ingenuity of its builders, who did not have sophisticated mathematical knowledge of structural forces or materials, is remarkable.

CH A P TE R 5

form and beauty

139

How buildings stay up ‘Engineers … have applied logical reasoning and mathematical methods in questions which have previously been settled by empirical formulae and rule of thumb.’

The Times engineering correspondent, 190821

In 1995, Channel 4 television aired a landmark series about structural engineering entitled The Limit. It explored the work of people pushing engineering to the boundaries of scientific knowledge, making longer bridges, deeper tunnels, larger aircraft, taller buildings, faster ships and space technology that could go farther than anything hitherto. And it raised an important point about how we see engineers and their practice.

Its producer, Martin Mortimore, commented on the place of engineers in today’s society compared with the well-known names from the past such as Thomas Telford, George Stephenson and Isambard Kingdom Brunel. ‘Somehow they seem to have been more in touch with their projects’, he remarked, ‘holding a slide rule in one hand and scribbling equations with the other.

Today, high-powered computer programs can solve many of the equations, and even come up with the designs’. However, he concluded, ‘there are still people of vision and imagination who can push back boundaries’.22 These comments were prescient. In the widest sense they were asking why we know more about nineteenth-century engineers than about those among us today. In part, this is owing to the way the history of engineering has been written in the past.

It has made heroes out of individuals, leaving the backroom women and men unnamed and under-appreciated. But Mortimore also raised an important point about the role of mathematics in the practice and identity of engineers. Are engineers mathematicians?

And how has the answer to this question shifted over the past few hundred years?

In the distant past, it is true to say that much engineering, particularly of large-scale structures and buildings, relied more

138

M a t h e ma t i c s

Sir Marc Isambard Brunel, c.1802 This portrait is typical of early nineteenth-century depictions of engineers, with an example of their work in the background and the instruments of engineering on drawings in the foreground. Those instruments – here, dividers and a set square – could equally belong to an architect, surveyor, navigator or mathematician. Marc Isambard Brunel was the father of Isambard Kingdom Brunel.

on rules of thumb. Sometimes this involved what we might today call over-engineering, or rather building with big margins of safety because the mathematics of the structure’s forces and materials, and the behaviour of the ground on which it was built, were not known well enough. At other times, buildings needed modification after construction to strengthen parts that were too weak having been under-engineered. This practical approach based on experience was empirical rather than theoretical. Some remarkable forms could be built in this way. Look at medieval cathedrals and chapels, with their soaring vaulted roofs and flying buttresses.

Their builders had no slide rules, but the structures were incredibly daring nonetheless.

But from the seventeenth century onwards, sweeping developments in science and mathematics put our understanding

of how structures behave – the forces, stresses and strains within them – onto a formal theoretical footing. Some of the most well-known names in science worked on what we now know as structural engineering, from Galileo Galilei and Isaac Newton to Robert Hooke, with Leonhard Euler, Claude-Louis Navier and Augustin-Louis Cauchy developing sophisticated mathematical theories still used extensively today. Instrument-makers developed similarly powerful tools, such as the slide rule invented in the 1620s. Taken together, this gave engineers both a theoretical and a practical mathematical toolkit to inform their practice.

So, in Telford’s day, and that of Stephenson and Brunel – all three flourished in the first half of the nineteenth century – the answer to the question ‘are engineers mathematicians’ would have been unquestionably ‘yes’. (They would also have called

The choir and vaulted roof of St George’s Chapel, Windsor, photographed c.1890 St George’s Chapel, Windsor, is one of the world’s finest examples of medieval architecture. Its almost flat vaulted roof is breathtaking, and the ingenuity of its builders, who did not have sophisticated mathematical knowledge of structural forces or materials, is remarkable.

CH A P TE R 5

form and beauty

139

We live in a world that threatens to overwhelm us. We live on vast continents bounded by fathomless oceans. We live in an infinite expanse of space surrounded by stars and planets performing mysterious dances across our skies. How can we possibly explore this world, let alone understand or even control it? The answer has always lain, in part, in mathematics, which we use to reduce our overwhelming world to a series of maps and models – physical and intellectual – at a scale that we can comprehend. In doing so, we reduce the world to quantity and order.

In astronomy, we have made miniature universes we can hold in our hands or keep in our homes. Astrolabes are two-dimensional versions of a threedimensional world, which we have used for a wide range of practical, intellectual and religious purposes. Astronomical clocks reproduce the motions of the heavenly bodies. Star charts and catalogues, made by gathering thousands of painstaking astronomical observations over many years, let us see patterns and understand our place in the universe. Geometry is the mathematical tool that has enabled this modelling of the universe.

Coming back down to Earth, as we have explored distant lands and built empires of places and people far away, the first task has been to make maps of what we have found. Maps are political artefacts that allow populations to be located and brought under control. It is no coincidence that Britain’s first official map project was the Ordnance Survey, run by the military department responsible for weaponry. Yet the same techniques of surveying, using the mathematics of trigonometry, have enabled us to map our own local areas and set out building sites.

Any nation with a coastline knows only too well the awesome power of the oceans. As we have explored and traded across the seas, we have tried to understand the actions of these vast bodies of water, which, driven by the Moon’s gravity and by the Earth’s violent weather system, can inundate our land and destroy communities. But by building models of the oceans’ movement we can at least predict disasters.

Detail from the astrolabe

by Jamal al-Din ibn Muqim, Lahore, c.1666 Object No: 1985-2077

146

M a t h e ma t i c s

Making maps and models is a remarkable human practice. It has enabled us to explore the distant reaches of our existence. Computers have transformed our ability to do this, not just by providing a technology that can be reshaped in an instant using software instructions rather than years of rebuilding or rewiring, but by allowing the modelling of intellectual ideas in mathematics that have no place in reality. This speculative, entirely abstract side of mathematical practice is an exploration in its own right, and means that mathematics will never be completed. There will always be more to discover.

CH A P TE R 6

M a PS a n d M OD e l S

147

We live in a world that threatens to overwhelm us. We live on vast continents bounded by fathomless oceans. We live in an infinite expanse of space surrounded by stars and planets performing mysterious dances across our skies. How can we possibly explore this world, let alone understand or even control it? The answer has always lain, in part, in mathematics, which we use to reduce our overwhelming world to a series of maps and models – physical and intellectual – at a scale that we can comprehend. In doing so, we reduce the world to quantity and order.

In astronomy, we have made miniature universes we can hold in our hands or keep in our homes. Astrolabes are two-dimensional versions of a threedimensional world, which we have used for a wide range of practical, intellectual and religious purposes. Astronomical clocks reproduce the motions of the heavenly bodies. Star charts and catalogues, made by gathering thousands of painstaking astronomical observations over many years, let us see patterns and understand our place in the universe. Geometry is the mathematical tool that has enabled this modelling of the universe.

Coming back down to Earth, as we have explored distant lands and built empires of places and people far away, the first task has been to make maps of what we have found. Maps are political artefacts that allow populations to be located and brought under control. It is no coincidence that Britain’s first official map project was the Ordnance Survey, run by the military department responsible for weaponry. Yet the same techniques of surveying, using the mathematics of trigonometry, have enabled us to map our own local areas and set out building sites.

Any nation with a coastline knows only too well the awesome power of the oceans. As we have explored and traded across the seas, we have tried to understand the actions of these vast bodies of water, which, driven by the Moon’s gravity and by the Earth’s violent weather system, can inundate our land and destroy communities. But by building models of the oceans’ movement we can at least predict disasters.

Detail from the astrolabe

by Jamal al-Din ibn Muqim, Lahore, c.1666 Object No: 1985-2077

146

M a t h e ma t i c s

Making maps and models is a remarkable human practice. It has enabled us to explore the distant reaches of our existence. Computers have transformed our ability to do this, not just by providing a technology that can be reshaped in an instant using software instructions rather than years of rebuilding or rewiring, but by allowing the modelling of intellectual ideas in mathematics that have no place in reality. This speculative, entirely abstract side of mathematical practice is an exploration in its own right, and means that mathematics will never be completed. There will always be more to discover.

CH A P TE R 6

M a PS a n d M OD e l S

147

Public and political fears about health risks from chemicals in the environment increased dramatically in 1962, when a former US government biologist, Rachel Carson, wrote the book Silent Spring, expressing concerns about the harmful and carcinogenic effects of chemical pesticides such as DDT, used throughout the world from the Second World War onwards to control malariacarrying mosquitoes and insect damage in crops. The effect of Silent Spring on the emerging environmental movement was powerful, as was the response of chemical companies, which heavily criticised the book’s findings.

Then, in the 1970s and 1980s, attention focused on the toxic chemical emissions from road vehicles, with growing concerns over long-term cancer-causing products in exhaust fumes from both petrol and diesel engines. Following government regulation in numerous countries, catalytic converters – exhaust pipe devices that turned some of the harmful pollutants into less harmful ones – became mandatory.

opposite Cabinet of tissue samples collected

by Chris Wagner, 1950–91 The pathologist Chris Wagner, together with two colleagues, published a paper in 1960 that first demonstrated a link between exposure to blue asbestos and the lung cancer mesothelioma. He collected samples from diseased patients over many years.

Object No: 2004-260

118

M a t h e ma t i c s

Canister of DDT insecticide spray, 1970 Fears over the possible health risk of toxic chemicals such as the insecticide DDT accelerated from 1962 with the publication of Rachel Carson’s powerful work, Silent Spring.

Object No: Y1995.2.210

above Roadside emissions monitoring equipment The UK’s Transport Research Laboratory carried out extensive tests on vehicle emissions, including air quality testing such as this roadside installation near London.

top Catalytic converter removed for testing in 1986 after 50,000 miles running Devices to reduce the emission of some toxic chemicals from vehicle exhaust pipes – known as catalytic converters – became mandatory on new cars in the UK in the 1990s following earlier moves in the USA. Object No: 1997-1822

CH A P TE R 4

l i f e a n d d e at h

119

Public and political fears about health risks from chemicals in the environment increased dramatically in 1962, when a former US government biologist, Rachel Carson, wrote the book Silent Spring, expressing concerns about the harmful and carcinogenic effects of chemical pesticides such as DDT, used throughout the world from the Second World War onwards to control malariacarrying mosquitoes and insect damage in crops. The effect of Silent Spring on the emerging environmental movement was powerful, as was the response of chemical companies, which heavily criticised the book’s findings.

Then, in the 1970s and 1980s, attention focused on the toxic chemical emissions from road vehicles, with growing concerns over long-term cancer-causing products in exhaust fumes from both petrol and diesel engines. Following government regulation in numerous countries, catalytic converters – exhaust pipe devices that turned some of the harmful pollutants into less harmful ones – became mandatory.

opposite Cabinet of tissue samples collected

by Chris Wagner, 1950–91 The pathologist Chris Wagner, together with two colleagues, published a paper in 1960 that first demonstrated a link between exposure to blue asbestos and the lung cancer mesothelioma. He collected samples from diseased patients over many years.

Object No: 2004-260

118

M a t h e ma t i c s

Canister of DDT insecticide spray, 1970 Fears over the possible health risk of toxic chemicals such as the insecticide DDT accelerated from 1962 with the publication of Rachel Carson’s powerful work, Silent Spring.

Object No: Y1995.2.210

above Roadside emissions monitoring equipment The UK’s Transport Research Laboratory carried out extensive tests on vehicle emissions, including air quality testing such as this roadside installation near London.

top Catalytic converter removed for testing in 1986 after 50,000 miles running Devices to reduce the emission of some toxic chemicals from vehicle exhaust pipes – known as catalytic converters – became mandatory on new cars in the UK in the 1990s following earlier moves in the USA. Object No: 1997-1822

CH A P TE R 4

l i f e a n d d e at h

119

The political upheaval of the early nineteenth century was part of a continuum, rather than a revolution. Relatively new technologies such as the modern theodolite – invented in the eighteenth century – simply brought prestige and accuracy to an old political practice. Today, theodolites are plain and plasticbodied, although more accurate and precise than ever before. They measure distances as well as angles and the results are stored in on-board data recorders for later use in computer software.

Surveying is an ancient mathematical activity that commanded the highest status. Certainly the surveyors of Ireland and India felt they were engaged in a prestigious and valuable project. Perhaps today we do not think of it as such.

But when we see people in high-visibility jackets operating theodolites on construction sites, we are witnessing the latest manifestation of a crucial mathematical practice and without it we would lose our way.

Total station theodolite with recording unit, c.1979 Many modern theodolites known as ‘total stations’, such as this one, measure distances as well as angles.

They also record measurements in on-board data recorders for later use in a computer.

Surveyor on the Crossrail project at Liverpool Street, London, 2012 Surveying using theodolites and other instruments has become a widespread practice, carried out by people as diverse as mapmakers, civil engineers, military officers and construction workers.

Object No: 1990-49/1

Index chart to the Great Trigonometrical Survey of India, 1870 This chart shows the triangles in the Survey of India, which took over 60 years to complete. Triangulation, using the mathematics of trigonometry, was a powerful tool of political control.

CH A P TE R 6

M a PS a n d M OD e l S

159

The political upheaval of the early nineteenth century was part of a continuum, rather than a revolution. Relatively new technologies such as the modern theodolite – invented in the eighteenth century – simply brought prestige and accuracy to an old political practice. Today, theodolites are plain and plasticbodied, although more accurate and precise than ever before. They measure distances as well as angles and the results are stored in on-board data recorders for later use in computer software.

Surveying is an ancient mathematical activity that commanded the highest status. Certainly the surveyors of Ireland and India felt they were engaged in a prestigious and valuable project. Perhaps today we do not think of it as such.

But when we see people in high-visibility jackets operating theodolites on construction sites, we are witnessing the latest manifestation of a crucial mathematical practice and without it we would lose our way.

Total station theodolite with recording unit, c.1979 Many modern theodolites known as ‘total stations’, such as this one, measure distances as well as angles.

They also record measurements in on-board data recorders for later use in a computer.

Surveyor on the Crossrail project at Liverpool Street, London, 2012 Surveying using theodolites and other instruments has become a widespread practice, carried out by people as diverse as mapmakers, civil engineers, military officers and construction workers.

Object No: 1990-49/1

Index chart to the Great Trigonometrical Survey of India, 1870 This chart shows the triangles in the Survey of India, which took over 60 years to complete. Triangulation, using the mathematics of trigonometry, was a powerful tool of political control.

CH A P TE R 6

M a PS a n d M OD e l S

159

Viewing the mathematical past: action, explanation, speculation Jim Bennett History Faculty, University of Oxford

are obliged to make some prior decisions, if they are to know what to write about or what to select for display. So, here is a pragmatic response to the conundrum of mathematical history: we will look for who identified themselves as mathematicians in the past and then ask how they justified that identity. We need not go as far as adapting to mathematics the claim for art, associated with Marcel Duchamp, that it is art if the artist says so, but we might transpose his more modest aphorism, ‘I don’t believe in art. I believe in artists’.

Looking across the population of those who were thought of, or presented themselves, as mathematicians since the Renaissance, we can note three very general approaches to the discipline. They are not mutually exclusive, as a mathematician might say, that is, they can, and in fact did, exist together. But the proposition here is that each, in turn, was the dominant approach among those who called themselves ‘mathematicians’. They can be labelled, in a chronological sequence, ‘action’, ‘explanation’ and ‘speculation’.

Action

How can there be a history of mathematics? At first glance, the question may seem foolish: surely mathematicians have a stronger sense of their subject’s past than those working in many scientific disciplines. Their heads are full of names of honoured predecessors, linked to principles, theorems and methods: Pythagoras’s theorem, Fermat’s last theorem, Fourier series. The mathematics of these earlier thinkers stretches back over thousands of years and across many cultures, but it seems immediate, vital, transparent and sometimes relevant to their successors today.

Historians, on the other hand, see men and women, the beliefs they hold and the principles they follow, shaped by their time, their culture and their individual lives. Historians are not expected to share their subjects’ convictions. The understanding they seek is more detached. They will identify the principles adopted by their subjects, analyse and interpret them, explain how they came to be accepted and how they were used, but make no judgement about their timeless truth.

The great majority of mathematical practice in the Renaissance and the early modern periods, perhaps up to the late seventeenth century, was understood as useful rather than illuminating: it was concerned with practical outcomes, not with understanding the natural world. Mathematical techniques were present in a growing range of mathematical arts and sciences, such as navigation, surveying, dialling, architecture, warfare, practical mechanics, cartography and perspective drawing. Even astronomy, treated geometrically, was not seen as providing an explanation of the motions of the heavens but was a tool for prediction that was especially useful in calendar regulation and astrology.

Who were the mathematicians in the era of action? There were a great many, of course, but we can name a few. Albrecht Dürer, now remembered mostly as a painter and printmaker, was a mathematician and published a substantial work on practical geometry. Gerard Mercator was a cartographer and mathematical instrument-maker, who used geometry to construct the most enduring projection of the world map. John Napier was a Scottish

Of course particular mathematical concerns, approaches, methods and languages come to characterise certain periods, regions and cultures. They take root in educational systems and are encouraged through commendation, appointments and awards. But for many mathematicians, underlying such diversity is a more fundamental, intellectual coherence, which is where the real mathematics is to be found, however it is expressed in a local context.

If the very doing of mathematics entails the assumption that its truths transcend time, place and culture, is it incompatible with the doing of history? This is not a matter of there being different branches of mathematics, operating on different assumptions, such as Euclidean and non-Euclidian geometries. Given the principles and definitions intrinsic to a domain of mathematics, mathematicians will still recognise matters as, in an uncompromised and independent sense, true or false. Is it, then, in the nature of mathematics to transcend its circumstances, so it has a past but not a history?

This short essay offers a structure for thinking about mathematics in the period covered by the Mathematics Gallery in the Science Museum. Writers of books and curators of galleries

174

M a t h e ma t i c s

Mathematics in action, 1622 A mariner’s astrolabe marks the centre of a geometrical diagram of the celestial sphere. The seaman measures the angle of the Sun above the horizon at noon and the diagram relates this measurement to finding latitude.

e s s ay s

175

Viewing the mathematical past: action, explanation, speculation Jim Bennett History Faculty, University of Oxford

are obliged to make some prior decisions, if they are to know what to write about or what to select for display. So, here is a pragmatic response to the conundrum of mathematical history: we will look for who identified themselves as mathematicians in the past and then ask how they justified that identity. We need not go as far as adapting to mathematics the claim for art, associated with Marcel Duchamp, that it is art if the artist says so, but we might transpose his more modest aphorism, ‘I don’t believe in art. I believe in artists’.

Looking across the population of those who were thought of, or presented themselves, as mathematicians since the Renaissance, we can note three very general approaches to the discipline. They are not mutually exclusive, as a mathematician might say, that is, they can, and in fact did, exist together. But the proposition here is that each, in turn, was the dominant approach among those who called themselves ‘mathematicians’. They can be labelled, in a chronological sequence, ‘action’, ‘explanation’ and ‘speculation’.

Action

How can there be a history of mathematics? At first glance, the question may seem foolish: surely mathematicians have a stronger sense of their subject’s past than those working in many scientific disciplines. Their heads are full of names of honoured predecessors, linked to principles, theorems and methods: Pythagoras’s theorem, Fermat’s last theorem, Fourier series. The mathematics of these earlier thinkers stretches back over thousands of years and across many cultures, but it seems immediate, vital, transparent and sometimes relevant to their successors today.

Historians, on the other hand, see men and women, the beliefs they hold and the principles they follow, shaped by their time, their culture and their individual lives. Historians are not expected to share their subjects’ convictions. The understanding they seek is more detached. They will identify the principles adopted by their subjects, analyse and interpret them, explain how they came to be accepted and how they were used, but make no judgement about their timeless truth.

The great majority of mathematical practice in the Renaissance and the early modern periods, perhaps up to the late seventeenth century, was understood as useful rather than illuminating: it was concerned with practical outcomes, not with understanding the natural world. Mathematical techniques were present in a growing range of mathematical arts and sciences, such as navigation, surveying, dialling, architecture, warfare, practical mechanics, cartography and perspective drawing. Even astronomy, treated geometrically, was not seen as providing an explanation of the motions of the heavens but was a tool for prediction that was especially useful in calendar regulation and astrology.

Who were the mathematicians in the era of action? There were a great many, of course, but we can name a few. Albrecht Dürer, now remembered mostly as a painter and printmaker, was a mathematician and published a substantial work on practical geometry. Gerard Mercator was a cartographer and mathematical instrument-maker, who used geometry to construct the most enduring projection of the world map. John Napier was a Scottish

Of course particular mathematical concerns, approaches, methods and languages come to characterise certain periods, regions and cultures. They take root in educational systems and are encouraged through commendation, appointments and awards. But for many mathematicians, underlying such diversity is a more fundamental, intellectual coherence, which is where the real mathematics is to be found, however it is expressed in a local context.

If the very doing of mathematics entails the assumption that its truths transcend time, place and culture, is it incompatible with the doing of history? This is not a matter of there being different branches of mathematics, operating on different assumptions, such as Euclidean and non-Euclidian geometries. Given the principles and definitions intrinsic to a domain of mathematics, mathematicians will still recognise matters as, in an uncompromised and independent sense, true or false. Is it, then, in the nature of mathematics to transcend its circumstances, so it has a past but not a history?

This short essay offers a structure for thinking about mathematics in the period covered by the Mathematics Gallery in the Science Museum. Writers of books and curators of galleries

174

M a t h e ma t i c s

Mathematics in action, 1622 A mariner’s astrolabe marks the centre of a geometrical diagram of the celestial sphere. The seaman measures the angle of the Sun above the horizon at noon and the diagram relates this measurement to finding latitude.

e s s ay s

175

Women who made mathematics count: Émilie du Châtelet and Mary Somerville Patricia Fara Department of History and Philosophy of Science, University of Cambridge

denied that it represented reality. Gradually the split narrowed until it was formally bridged in 1687, when Isaac Newton published his Latin Principia, the ‘Mathematical Principles of Natural Philosophy’. This carefully worded title declares that the world can be both described and explained mathematically.

Newton is now universally celebrated as one of the greatest scientific geniuses that ever lived, but he owes much of his fame to admirers who clarified his mathematics and consolidated his reputation. Newton himself was an appalling lecturer who had little interest in accessibility and left much of his work unpublished, so that his almost mythological stature was largely built up after his death. Even those who knew nothing about equations learnt about his reputation through pictures, articles and also poetry, a far more popular genre then than now. The night that Newton died, poet Elizabeth Tollet eulogised his innate brilliance: Thy Mind, enlarg’d by nature to compute Her vastest Work, cou’d trace the most minute.2

Mathematical puzzles are now a standard feature in colour supplements, but they first appeared in an unexpected place: a women’s magazine. In 1709, the male editor of the annual Ladies’ Diary announced that, by popular request, recipes would be replaced by ‘enigmas, and arithmetical questions, [which] give the greatest satisfaction and delight to the obliging fair.’1 Every year, answers flooded in from women who could never go to university, but apparently preferred the challenge of mathematics to conventional activities such as flower-arranging or embroidery.

Women rarely feature in books about the history of mathematics. The most obvious explanation for this scarcity is their lack of educational opportunities. But there is another reason: historians generally focus on men who lent their names to famous theorems, rather than describing how and why mathematics has grown in significance over the centuries so that it is now crucial for modern science, the lynchpin of our technological society.

Even the greatest of mathematicians would seem irrelevant if there were no publications communicating their discoveries to their colleagues, no textbooks explaining their achievements to subsequent generations, and no translations ensuring the international spread of their ideas. Such mathematical projects were enormously influential – and many of them were initiated by women.

Although women were excluded from universities, a few were sufficiently rich, intelligent and independent to hire private tutors and study at home. By interpreting, criticising and synthesising new discoveries they contributed to the collective endeavour that made mathematics vitally important throughout the world. Mathematics is now central to science and society, but 500 years ago the major sources of learning were Aristotle and the Bible, neither of which had much to say about mathematics. Although studying the natural world gradually became accepted as a route to knowledge, the traditional divide remained between natural philosophers who looked for causes and tried to explain why things happen, and mathematicians who modelled the world and described how things happen. When Nicolaus Copernicus suggested that the Sun might lie at the centre of the universe, many of his colleagues praised the mathematical clarity of his scheme, but

178

M a t h e ma t i c s

At one time Newton’s neighbour in the Tower of London, Tollet seems to be one of the few women with whom Newton actually conversed. A child prodigy forced to stay at home while her two younger brothers drunk and gambled their way through Oxbridge, she was unable to develop her mathematical talents and could contribute to Newton’s swelling prestige only through verse.

Other talented women were more fortunate than Tollet. Even though their lives were severely restricted by modern standards, they were able to continue their studies to an advanced level. Two in particular were crucial in the establishment of the Newtonian system as it is now understood: Émilie du Châtelet, who helped to transport Newtonian ideas across the Channel in the mid eighteenth century, and Mary Somerville, a Scottish mathematician who imported a French modification of Newtonianism back into Britain around a hundred years later. Émilie du Châtelet, wrote the French playwright and essayist Voltaire, ‘was a great man whose only fault was being a woman.’3 Trapped between the sexes, du Châtelet conformed to social norms by shopping, dancing and entertaining with great enthusiasm (an astute mathematician, she subsidised some of her pleasures by success at the gambling table).

On the other hand, she behaved abnormally by dedicating herself to Newtonian natural philosophy, writing, experimenting and plunging her hands into ice-cold water to keep herself awake when deadlines were close. Despite this unconventional lifestyle, she still fulfilled the traditional time-filling duties demanded from a wife and mother.

Born into a wealthy aristocratic family, du Châtelet benefited from an enlightened father who gave her an education more typical for a boy than a girl, but it was not until her late 20s that she began to immerse herself in Newtonian ideas. Married to an older army officer and with two surviving children, she persuaded one of several admirers to teach her mathematics privately, and fell passionately in love with another, Voltaire. Because he was being pursued by the authorities for his radical political views, du Châtelet sent him into hiding at her country estate. The next summer she joined him there, and for 15 years they lived together, enmeshed in a passionate affair laced with intense intellectual activity.

France was still dominated by the physics and mathematics of its own national hero, René Descartes, but in 1738 an introductory book on Newtonian philosophy convinced the nation to switch its allegiance. Well illustrated, it clearly explained the basic principles of Newton’s discoveries, so that for the first time the new mathematical physics became accessible to a wide range of French people. Although only Voltaire’s name is on the title page of their

e s s ay s

179

Women who made mathematics count: Émilie du Châtelet and Mary Somerville Patricia Fara Department of History and Philosophy of Science, University of Cambridge

denied that it represented reality. Gradually the split narrowed until it was formally bridged in 1687, when Isaac Newton published his Latin Principia, the ‘Mathematical Principles of Natural Philosophy’. This carefully worded title declares that the world can be both described and explained mathematically.

Newton is now universally celebrated as one of the greatest scientific geniuses that ever lived, but he owes much of his fame to admirers who clarified his mathematics and consolidated his reputation. Newton himself was an appalling lecturer who had little interest in accessibility and left much of his work unpublished, so that his almost mythological stature was largely built up after his death. Even those who knew nothing about equations learnt about his reputation through pictures, articles and also poetry, a far more popular genre then than now. The night that Newton died, poet Elizabeth Tollet eulogised his innate brilliance: Thy Mind, enlarg’d by nature to compute Her vastest Work, cou’d trace the most minute.2

Mathematical puzzles are now a standard feature in colour supplements, but they first appeared in an unexpected place: a women’s magazine. In 1709, the male editor of the annual Ladies’ Diary announced that, by popular request, recipes would be replaced by ‘enigmas, and arithmetical questions, [which] give the greatest satisfaction and delight to the obliging fair.’1 Every year, answers flooded in from women who could never go to university, but apparently preferred the challenge of mathematics to conventional activities such as flower-arranging or embroidery.

Women rarely feature in books about the history of mathematics. The most obvious explanation for this scarcity is their lack of educational opportunities. But there is another reason: historians generally focus on men who lent their names to famous theorems, rather than describing how and why mathematics has grown in significance over the centuries so that it is now crucial for modern science, the lynchpin of our technological society.

Even the greatest of mathematicians would seem irrelevant if there were no publications communicating their discoveries to their colleagues, no textbooks explaining their achievements to subsequent generations, and no translations ensuring the international spread of their ideas. Such mathematical projects were enormously influential – and many of them were initiated by women.

Although women were excluded from universities, a few were sufficiently rich, intelligent and independent to hire private tutors and study at home. By interpreting, criticising and synthesising new discoveries they contributed to the collective endeavour that made mathematics vitally important throughout the world. Mathematics is now central to science and society, but 500 years ago the major sources of learning were Aristotle and the Bible, neither of which had much to say about mathematics. Although studying the natural world gradually became accepted as a route to knowledge, the traditional divide remained between natural philosophers who looked for causes and tried to explain why things happen, and mathematicians who modelled the world and described how things happen. When Nicolaus Copernicus suggested that the Sun might lie at the centre of the universe, many of his colleagues praised the mathematical clarity of his scheme, but

178

M a t h e ma t i c s

At one time Newton’s neighbour in the Tower of London, Tollet seems to be one of the few women with whom Newton actually conversed. A child prodigy forced to stay at home while her two younger brothers drunk and gambled their way through Oxbridge, she was unable to develop her mathematical talents and could contribute to Newton’s swelling prestige only through verse.

Other talented women were more fortunate than Tollet. Even though their lives were severely restricted by modern standards, they were able to continue their studies to an advanced level. Two in particular were crucial in the establishment of the Newtonian system as it is now understood: Émilie du Châtelet, who helped to transport Newtonian ideas across the Channel in the mid eighteenth century, and Mary Somerville, a Scottish mathematician who imported a French modification of Newtonianism back into Britain around a hundred years later. Émilie du Châtelet, wrote the French playwright and essayist Voltaire, ‘was a great man whose only fault was being a woman.’3 Trapped between the sexes, du Châtelet conformed to social norms by shopping, dancing and entertaining with great enthusiasm (an astute mathematician, she subsidised some of her pleasures by success at the gambling table).

On the other hand, she behaved abnormally by dedicating herself to Newtonian natural philosophy, writing, experimenting and plunging her hands into ice-cold water to keep herself awake when deadlines were close. Despite this unconventional lifestyle, she still fulfilled the traditional time-filling duties demanded from a wife and mother.

Born into a wealthy aristocratic family, du Châtelet benefited from an enlightened father who gave her an education more typical for a boy than a girl, but it was not until her late 20s that she began to immerse herself in Newtonian ideas. Married to an older army officer and with two surviving children, she persuaded one of several admirers to teach her mathematics privately, and fell passionately in love with another, Voltaire. Because he was being pursued by the authorities for his radical political views, du Châtelet sent him into hiding at her country estate. The next summer she joined him there, and for 15 years they lived together, enmeshed in a passionate affair laced with intense intellectual activity.

France was still dominated by the physics and mathematics of its own national hero, René Descartes, but in 1738 an introductory book on Newtonian philosophy convinced the nation to switch its allegiance. Well illustrated, it clearly explained the basic principles of Newton’s discoveries, so that for the first time the new mathematical physics became accessible to a wide range of French people. Although only Voltaire’s name is on the title page of their

e s s ay s

179

Mathematics: a living, changing landscape Dame Celia Hoyles UCL Institute of Education, University College London Helen Wilson Department of Mathematics, University College London

‘Seniora Fawcett’, 1890 Philippa Fawcett was celebrated in Punch, 21 June 1890.

Hail the triumph of the corset, Hail the fair Philippa Fawcett, Victress in the fray Crown her queen of hydrostatics And the other mathematics Wreathe her brow in bay.11 Both Charlotte Scott and Philippa Fawcett were born into enlightened families who encouraged their mathematical endeavours. Charlotte’s father was a minister in the Congregational Church, a church known for its support of women’s rights; Philippa’s father had been a seventh wrangler and her mother, by her own admission, was ‘a Woman’s suffragist from the cradle’.12 While such families were in the minority, they were not alone.

Between 1880 and 1900, over 250 women successfully passed the Mathematical Tripos examination. However, teaching aside, there were few opportunities for these women to use the mathematics they had learnt. Charlotte Scott, unable to make a career in Britain, moved to America in 1885 to take up a position as one (and the only female) of the founding professors at Bryn Mawr College, Pennsylvania. She was a popular teacher and garnered acclaim for her research in algebraic geometry. Philippa Fawcett remained in mathematics as a lecturer at Newnham for a few years before eventually moving into education.

The achievements of both Charlotte Scott and Philippa Fawcett did much to further the cause of women’s rights in the nineteenth century but despite several hard-fought campaigns with an increasing number of men rallying to the cause, it was not enough to persuade Cambridge University to change its stance on awarding women degrees. An MA entitled the holder to voting rights within the university, and for too many men it was too high a price to pay. Two world wars would have to be fought and won before, in 1948, women would be entitled to degrees at Cambridge.

186 186

M Ma at th he e ma ma t t ii c cs s

Mathematics has a problem: a problem of perception. In the Channel 4 series Educating Cardiff (2015), a pupil grumbles, ‘When are you ever gonna need Pythagoras in your life?’ Many will sympathise with that feeling. To most people, mathematics is simply a collection of content fragments and calculations, known for centuries: a mountain to be climbed. There is little idea about the very essence of mathematics: its structure, logical framework and network of interrelated concepts, its ever-changing frontiers. This leads almost inevitably to ignorance of why on Earth mathematics might be important.

Worse: over the years, students have been subconsciously absorbing the message that mathematics is ‘not for people like me’. A 1966 poster entitled ‘Men of Modern Mathematics’, produced for IBM for display in classrooms, presents an image of isolated, largely white male figures who all look rather distinguished and superior, not a community that most would be comfortable to join. It would not be just girls who were put off. Things have changed since 1966, as indicated in the 2014 picture of children being inspired by mathematics in the Science Museum’s Pattern Pod hands-on gallery.

The community of mathematicians in 2015 is much more diverse: the London Mathematical Society now runs a twitter feed @womeninmaths with over 2000 followers in 2015 and rapidly growing; one of the top mathematics prizes, the Fields Medal, was awarded for the first time in 2014 to a female mathematician, Maryam Mirzakhani; and girls make up roughly half of A-Level mathematics candidates and of mathematics undergraduates.

There is widespread acceptance that mathematics is important for everybody and for society. In most countries, the high status of mathematics and mathematics education is rarely contested, not least as mathematics is central for the health of the economy.1 But mathematics is so much more than this and to change perceptions of the subject we must work on a broader canvas than simply extrinsic reward.

Mathematics is practical. Charles Babbage, back in the 1840s, studying the costs of mail delivery, sowed the seeds of the mathematical discipline of Operational Research: a method of looking for the best solution to a problem where there are many valid solutions.

t rad e a n d t ra v e ls s a y s

187

Mathematics: a living, changing landscape Dame Celia Hoyles UCL Institute of Education, University College London Helen Wilson Department of Mathematics, University College London

‘Seniora Fawcett’, 1890 Philippa Fawcett was celebrated in Punch, 21 June 1890.

Hail the triumph of the corset, Hail the fair Philippa Fawcett, Victress in the fray Crown her queen of hydrostatics And the other mathematics Wreathe her brow in bay.11 Both Charlotte Scott and Philippa Fawcett were born into enlightened families who encouraged their mathematical endeavours. Charlotte’s father was a minister in the Congregational Church, a church known for its support of women’s rights; Philippa’s father had been a seventh wrangler and her mother, by her own admission, was ‘a Woman’s suffragist from the cradle’.12 While such families were in the minority, they were not alone.

Between 1880 and 1900, over 250 women successfully passed the Mathematical Tripos examination. However, teaching aside, there were few opportunities for these women to use the mathematics they had learnt. Charlotte Scott, unable to make a career in Britain, moved to America in 1885 to take up a position as one (and the only female) of the founding professors at Bryn Mawr College, Pennsylvania. She was a popular teacher and garnered acclaim for her research in algebraic geometry. Philippa Fawcett remained in mathematics as a lecturer at Newnham for a few years before eventually moving into education.

The achievements of both Charlotte Scott and Philippa Fawcett did much to further the cause of women’s rights in the nineteenth century but despite several hard-fought campaigns with an increasing number of men rallying to the cause, it was not enough to persuade Cambridge University to change its stance on awarding women degrees. An MA entitled the holder to voting rights within the university, and for too many men it was too high a price to pay. Two world wars would have to be fought and won before, in 1948, women would be entitled to degrees at Cambridge.

186 186

M Ma at th he e ma ma t t ii c cs s

Mathematics has a problem: a problem of perception. In the Channel 4 series Educating Cardiff (2015), a pupil grumbles, ‘When are you ever gonna need Pythagoras in your life?’ Many will sympathise with that feeling. To most people, mathematics is simply a collection of content fragments and calculations, known for centuries: a mountain to be climbed. There is little idea about the very essence of mathematics: its structure, logical framework and network of interrelated concepts, its ever-changing frontiers. This leads almost inevitably to ignorance of why on Earth mathematics might be important.

Worse: over the years, students have been subconsciously absorbing the message that mathematics is ‘not for people like me’. A 1966 poster entitled ‘Men of Modern Mathematics’, produced for IBM for display in classrooms, presents an image of isolated, largely white male figures who all look rather distinguished and superior, not a community that most would be comfortable to join. It would not be just girls who were put off. Things have changed since 1966, as indicated in the 2014 picture of children being inspired by mathematics in the Science Museum’s Pattern Pod hands-on gallery.

The community of mathematicians in 2015 is much more diverse: the London Mathematical Society now runs a twitter feed @womeninmaths with over 2000 followers in 2015 and rapidly growing; one of the top mathematics prizes, the Fields Medal, was awarded for the first time in 2014 to a female mathematician, Maryam Mirzakhani; and girls make up roughly half of A-Level mathematics candidates and of mathematics undergraduates.

There is widespread acceptance that mathematics is important for everybody and for society. In most countries, the high status of mathematics and mathematics education is rarely contested, not least as mathematics is central for the health of the economy.1 But mathematics is so much more than this and to change perceptions of the subject we must work on a broader canvas than simply extrinsic reward.

Mathematics is practical. Charles Babbage, back in the 1840s, studying the costs of mail delivery, sowed the seeds of the mathematical discipline of Operational Research: a method of looking for the best solution to a problem where there are many valid solutions.

t rad e a n d t ra v e ls s a y s

187

In memory of Dame Zaha Hadid 1950 – 2016

Produced exclusively for SCMG Enterprises Ltd by Scala Arts & Heritage Publishers in conjunction with Mathematics: The Winton Gallery at the Science Museum, London.

First published in 2016 by Scala Arts & Heritage Publishers Ltd 10 Lion Yard Tremadoc Road London SW4 7NQ, UK www.scalapublishers.com In association with Science Museum Exhibition Road London SW7 2DD www.sciencemuseum.org.uk Every purchase supports the Museum.

Texts © SCMG Enterprises Ltd, 2016 Science Museum ® SCMG Enterprises Ltd and designs © SCMG Enterprises Ltd This edition © Scala Arts & Heritage Publishers Ltd, 2016 Project manager and copy editor: Linda Schofield Designer: Will Webb Design Printed and bound in China

ISBN 978-1-78551-039-7 (hardback) ISBN 978-1-78551-049-6 (paperback) 10 9 8 7 6 5 4 3 2 1 All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise, without the written permission of Scala Arts & Heritage Publishers Ltd and SCMG Enterprises Ltd. Every effort has been made to acknowledge correct copyright of images where applicable. Any errors or omissions are unintentional and should be notified to the Publisher, who will arrange for corrections to appear in any reprints.

British Library Cataloguing in Publication Data.

A catalogue record for this book is available from the British Library.

Frontispiece: Detail from calculator for problems of reinforced concrete, 1920s (see p.142).

Front and back cover: Set of glass shapes for teaching elementary geometry, c.1800 (see p.149).

Endpapers/inside cover and chapter openers: taken from J G Heck, Iconographic Encyclopaedia of Science, Literature, and Art, Volume 1 (plates) (New York: Rudolph Garrigue, 1851).