Tyrants and geometers

Reading Proof!: How the World Became Geometrical by Amir Alexander (Scientific American) for the Telegraph, 7 November 2019

The fall from grace of Nicolas Fouquet, Louis XIV’s superintendant of finances, was spectacular and swift. In 1661 he held a fete to welcome the king to his gardens at Vaux-le-Vicomte. The affair was meant to flatter, but its sumptuousness only served to convince the absolutist monarch that Fouquet was angling for power. “On 17 August, at six in the evening Fouquet was the King of France,” Voltaire observed; “at two in the morning he was nobody.”

Soon afterwards, Fouquet’s gardens were grubbed up in an act, not of vandalism, but of expropriation: “The king’s men carefully packed the objects into crates and hauled them away to a marshy town where Louis was intent on building his own dream palace,” the Israeli-born US historian Amir Alexander tells us. “It was called Versailles.”

Proof! explains how French formal gardens reflected, maintained and even disseminated the political ideologies of French monarchs. from “the Affable” Charles VIII in the 15th century to poor doomed Louis XVI, destined for the guillotine in 1793. Alexander claims these gardens were the concrete and eloquent expression of the idea that “geometry was everywhere and structured everything — from physical nature to human society, the state, and the world.”

If you think geometrical figures are abstract artefacts of the human mind, think again. Their regularities turn up in the natural world time and again, leading classical thinkers to hope that “underlying the boisterous chaos and variety that we see around us there may yet be a rational order, which humans can comprehend and even imitate.”

It is hard for us now to read celebrations of nature into the rigid designs of 16th century Fontainebleau or the Tuileries, but we have no problem reading them as expressions of political power. Geometers are a tyrant’s natural darlings. Euclid spent many a happy year in Ptolemaic Egypt. King Hiero II of Syracuse looked out for Archimedes. Geometers were ideologically useful figures, since the truths they uncovered were static and hierarchical. In the Republic, Plato extols the virtues of geometry and advocates for rigid class politics in practically the same breath.

It is not entirely clear, however, how effective these patterns actually were as political symbols. Even as Thomas Hobbes was modishly emulating the logical structure of Euclid’s (geometrical) Elements in the composition of his (political) Leviathan (demonstrating, from first principles, the need for monarchy), the Duc de Saint-Simon, a courtier and diarist, was having a thoroughly miserable time of it in the gardens of Louis XIV’s Versailles: “the violence everywhere done to nature repels and wearies us despite ourselves,” he wrote in his diary.

So not everyone was convinced that Versailles, and gardens of that ilk, revealed the inner secrets of nature.

Of the strictures of classical architecture and design, Alexander comments that today, “these prescriptions seem entirely arbitrary”. I’m not sure that’s right. Classical art and architecture is beautiful, not merely for its antiquity, but for the provoking way it toys with the mechanics of visual perception. The golden mean isn’t “arbitrary”.

It was fetishized, though: Alexander’s dead right about that. For centuries, Versailles was the ideal to which Europe’s grand urban projects aspired, and colonial new-builds could and did out-do Versailles, at least in scale. Of the work of Lutyens and Baker in their plans for the creation of New Delhi, Alexander writes: “The rigid triangles, hexagons, and octagons created a fixed, unalterable and permanent order that could not be tampered with.”

He’s setting colonialist Europe up for a fall: that much is obvious. Even as New Delhi and Saigon’s Boulevard Norodom and all the rest were being erected, back in Europe mathematicians Janos Bolyai, Carl Friedrich Gauss and Bernhard Riemann were uncovering new kinds of geometry to describe any curved surface, and higher dimensions of any order. Suddenly the rigid, hierarchical order of the Euclidean universe was just one system among many, and Versailles and its forerunners went from being diagrams of cosmic order to being grand days out with the kids.

Well, Alexander needs an ending, and this is as good a place as any to conclude his entertaining, enlightening, and admirably well-focused introduction to a field of study that, quite frankly, is more rabbit-hole than grass.

I was in Washington the other day, sweating my way up to the Lincoln Memorial. From the top I measured the distance, past the needle of the Washington Monument, to Capitol Hill. Major Pierre Charles L’Enfant built all this: it’s a quintessential product of the Versailles tradition. Alexander calls it “nothing less than the Constitutional power structure of the United States set in stone, pavement, trees, and shrubs.”

For nigh-on 250 years tourists have been slogging from one end of the National Mall to the other, re-enacting the passion of the poor Duc de Saint-Simon in Versailles, who complained that “you are introduced to the freshness of the shade only by a vast torrid zone, at the end of which there is nothing for you but to mount or descend.”

Not any more, though. Skipping down the steps, I boarded a bright red electric Uber scooter and sailed electrically east toward Capitol Hill. The whole dignity-dissolving charade was made possible (and cheap) by map-making algorithms performing geometrical calculations that Euclid himself would have recognised. Because the ancient geometer’s influence on our streets and buildings hasn’t really vanished. It’s been virtualised. Algorithmized. Turned into a utility.

Now geometry’s back where it started: just one more invisible natural good.

Eugenic America: how to exclude almost everyone

imbeciles

Imbeciles: The Supreme Court, American eugenics, and the sterilization of Carrie Buck by Adam Cohen (Penguin Press)

Defectives in the Land: Disability and immigration in the age of eugenics by Douglas C. Baynton (University of Chicago Press)

for New Scientist, 22 March 2016

ONE of 19th-century England’s last independent “gentleman scientists”, Francis Galton was the proud inventor of underwater reading glasses, an egg-timer-based speedometer for cyclists, and a self-tipping top hat. He was also an early advocate of eugenics, and his Hereditary Genius was published two years after the first part of Karl Marx’s Das Kapital.

Both books are about the betterment of the human race: Marx supposed environment was everything; Galton assumed the same for heredity. “If a twentieth part of the cost and pains were spent in measures for the improvement of the human race that is spent on the improvement of the breed of horses and cattle,” he wrote, “what a galaxy of genius might we not create! We might introduce prophets and high priests of civilisation into the world, as surely as we… propagate idiots by mating cretins.”

What would such a human breeding programme look like? Would it use education to promote couplings that produced genetically healthy offspring? Or would it discourage or prevent pairings that would otherwise spread disease or dysfunction? And would it work by persuasion or by compulsion?

The study of what was then called degeneracy fell to a New York social reformer, Richard Louis Dugdale. During an 1874 inspection of a jail in New York State, Dugdale learned that six of the prisoners there were related. He traced the Jukes family tree back six generations, and found that some 350 people related to this family by blood or marriage were criminals, prostitutes or destitute.

Dugdale concluded that, like genius, “degeneracy” runs in families, but his response was measured. “The licentious parent makes an example which greatly aids in fixing habits of debauchery in the child. The correction,” he wrote, “is change of the environment… Where the environment changes in youth, the characteristics of heredity may be measurably altered.”

Other reformers were not so circumspect. An Indiana reformatory promptly launched a eugenic sterilisation effort, and in 1907 Indiana enacted the world’s first compulsory sterilisation statute. California followed suit in 1909. Between 1927 and 1979, Virginia forcibly sterilised at least 7450 “unfit” people. One of them was Carrie Buck, a woman labelled feeble-minded and kept ignorant of the details of her own case right up to the point in October 1927 when her fallopian tubes were tied and cauterised using carbolic acid and alcohol.

In Imbeciles, Adam Cohen follows Carrie Buck through the US court system, past the desks of one legal celebrity after the other, and not one of them, not Howard Taft, not Louis Brandeis, not Oliver Wendell Holmes Jr, gave a damn about her.

Cohen anatomises in pitiless detail how inept civil society can be at assimilating scientific ideas. He also does a good job explaining why attempts to manipulate the genetic make-up of whole populations can only fail to improve the genetic health of our species. Eugenics fails because it looks for genetic solutions to what are essentially cultural problems. The anarchist biologist Peter Kropotkin made this point as far back as 1912. Who were unfit, he asked the first international eugenics congress in London: workers or monied idlers? Those who produced degenerates in slums or those who produced degenerates in palaces? Culture casts a huge influence over the way we live our lives, hopelessly complicating our measures of strength, fitness and success.

Readers of Cohen’s book would also do well to watch out for Douglas Baynton’s Defectives in the Land, to be published in June. Focusing on immigrant experiences in New York, Baynton explains how ideas about genetics, disability, race, family life and employment worked together to exclude an extraordinarily diverse range of men and women from the shores of the US.

“Doesn’t this squashy sentimentality of a big minority of our people about human life make you puke?” Holmes once exclaimed. Holmes was a miserable bigot, but he wasn’t wrong to thirst for more rigour in our public discourse. History is not kind to bad ideas.

“What of it, let them die!” – Sergei Oldenburg in Moscow

Baby steps towards an anecdotal history of Russian science…

Pomogi

Sergei Fyodorovich Oldenburg, secretary, Academy of Sciences (1863– 1934)

On 13 July 1921, Maxim Gorky appealed to the world for help; less than three months later later, Frank Golder, a native of Odessa with a PhD from Harvard University, found himself steeped in the horrors of the great Russian famine.

Golder had come from Washington to survey the extent of the catastrophe for Herbert Hoover’s American Relief Administration. His reports painted a terrible and complex picture. Russian agriculture had been virtually wiped out by a world war, a revolution, a civil war and then, in 1921, a drought. A government scheme to redistribute food had further alienated Russia’s traditionally suspicious peasant class; many buried and even burned their crops, sooner than hand them over to the Red Army. A survey team member wrote: “There were abandoned homes in the communes by the score, the roofs and wooden parts taken off for fuel, and the walls of mud and straw falling into decay. Everywhere we found emaciated starving children, with stomachs distended from eating melon rinds, cabbage leaves and anything that could be found, things which filled the stomach but did not nourish…”

Arriving in Penza, south-east of Moscow, Golder found the town stricken with cholera and typhus. There were next to no medicines. An 800- bed hospital there had only two thermometers, and the administrator’s best assistant, “thoroughly discouraged”, had committed suicide the day before.

In Moscow, things were better, but even among the reasonably well provided-for members of the Russian Academy of Sciences, 21 had died from disease and malnutrition.

Golder got the impression that Sergei Oldenburg, the Permanent Secretary of the Academy, would soon be joining them. “It is so pitiful and so heart breaking that it completely upsets me,” Golder wrote, recalling his visit for his friend and patron Ephraim D. Adams. “I wish you could meet him for he is one of the most scholarly, cultured and kindly men that I have ever met.” Oldenburg, nearly 60 by then, was bedridden. He was barely able to reach the scraps of toast on his table, let alone chew them. He had spent the past four years trying to support his own family, the orphan children of his brother, his daughter-in-law and his grandchildren, all on about nine million roubles – or five US dollars – a month. The children had bread every day: for the rest of the family, it was a weekly treat.

Sergei Oldenburg was one of the privileged ones: highly-educated, patrician, a personal acquaintance of Vladimir Lenin, and engaged in work vital to the state. Along with other scholars, he had even been recieving a dole, although, as he wrote, “it needed the limitless authority of Lenin and the enormous popularity of Gorky to carry off the issuing of an ‘academic ration’. For this exceptional ration was created before the eyes of the hungry masses who had set themselves the task of destroying all privileges and hierarchies.”

Class resentment, exacerbated by the emergency, was indeed fierce: Frank Golder recalls how one professor called upon the representative of the Crimean government – a young female Communist – to point out that professors in the Crimea were dying of hunger. “What of it?” she had replied. “Let them die!”

Since the October Revolution, Sergei Oldenburg had worked “like a giant” trying to keep up the Academy, trying to find the Academicians something to eat, trying to keep on good terms with the Bolsheviks while striving not to alienate the anti-Bolsheviks. Oldenburg, a world-renowned Orientalist, grandson of a Full General in the Imperial Russian Army, and with a modest amount of blue blood running through his veins, was a liberal nationalist; never a communist. In 1905 he had served in the Russian Provisional Government as Minister of Education. Unlike his political colleagues, however, he had chosen to remain in Russia following the Bolshevik takeover. When Golder, seated at Oldenburg’s bedside in Moscow, talked strenuously about American state recognition, and the good American investment capital might do to save the country, Oldenburg’s mixture of national pride, and his precoccuption with the redemptive powers of suffering, marked him as a leftover of a bygone age:

“Our salvation can not come from without but must come from within and we, as a government and to some extent as a nation have not yet confessed and repented our sins… Let us recover slowly, let us suffer some more the cruel pangs of hunger because it is the only way to get well and strong… All the suffering, all the misery we have endured and are enduring is teaching us Russians to think clearly and that is a great step in the line of progress.”

Oldenburg was certainly a clear thinker: a scholar of Buddhism who welcomed and enjoyed the company of the growing number of Academicians who were natural scientists. The Academy itself was old, founded in 1724 by Peter the Great. It had always been a more reliable friend to the State than the universities, and had enjoyed a privileged position, as a sort of expert arm of the Russian civil service. Oldenburg knew how to convey the Academy’s value to those now in power. He emphasised the practical benefits working with the Academy. His close friend Vladimir Vernadsky had established the Academy’s Commission for the Study of Natural Productive Forces (KEPS) – a key asset in negotiations with the government. Oldenburg asked for money and independence; in return, he could help the State develop greater self-sufficiency in raw materials and manufactures, and even help Lenin with his over-ambitious plan for the rapid electrification of Russia.
It was never an easy compromise, but Lenin, for his part, understood how important the Academy was to the Russia’s survival. (When, in 1922, a Proletkult bigwig wrote a Pravda article hostile to the Academy, Lenin, unimpressed, scrawled in the margin: “And what percentage of [his] loyal proliterians know how to build locomotives?”) So Sergei Oldenburg survived the famine, the flood that inundated his apartment in 1924, and even the attentions of “that black cloud from Moscow”, the astronomer Vartan Ter-Oganezov, an ideologue whose ambitions to remake science in the image of Bolshevism earn him a chapter later in this account. In this chapter, we will see how Oldenburg, with astonishing political dexterity, shaped the future of the world’s largest scientific institution: a sprawling organisation that fed and clothed almost all the people whose lives and and careers are described in this book.

From the beginning, Russian scientists had reservations about communist ideology. Until the “Great Break” and Cultural Revolution of 1929 there was not one member of the Academy of Sciences who was also a member of the Communist Party. But few Academicians could resist the allure of Sergei Oldenburg’s vision of the the Academy’s future: a scientistic programme of modernisation that offered many influential positions to the scientists and engineers willing to work with the communist government. The new Academy grew vast, comprising hundreds of research institutes spread across the USSR. Its central control structure appealed to Lenin’s notorious successor Joseph Stalin. But it appealed just as much to Academicians of Oldenburg’s stripe and generation: men who, in Tsarist times. had argued for nothing else.
Class resentment, wielded as a weapon by Joseph Stalin, eventually destroyed the arrangements Oldenburg spent so many years maintaining. Oldenburg was dismissed from his post during the “Great Break” of 1929. His legacy lived on, nonetheless: a collosal working institution, often troubled, often compromised, but recognised the world over as a pillar of world science.

“What of it, let them die!” – Sergei Oldenburg in Moscow

Baby steps towards an anecdotal history of Russian science…

Pomogi

Sergei Fyodorovich Oldenburg, secretary, Academy of Sciences (1863– 1934)

On 13 July 1921, Maxim Gorky appealed to the world for help; less than three months later later, Frank Golder, a native of Odessa with a PhD from Harvard University, found himself steeped in the horrors of the great Russian famine.

Golder had come from Washington to survey the extent of the catastrophe for Herbert Hoover’s American Relief Administration. His reports painted a terrible and complex picture. Russian agriculture had been virtually wiped out by a world war, a revolution, a civil war and then, in 1921, a drought. A government scheme to redistribute food had further alienated Russia’s traditionally suspicious peasant class; many buried and even burned their crops, sooner than hand them over to the Red Army. A survey team member wrote: “There were abandoned homes in the communes by the score, the roofs and wooden parts taken off for fuel, and the walls of mud and straw falling into decay. Everywhere we found emaciated starving children, with stomachs distended from eating melon rinds, cabbage leaves and anything that could be found, things which filled the stomach but did not nourish…”

Arriving in Penza, south-east of Moscow, Golder found the town stricken with cholera and typhus. There were next to no medicines. An 800- bed hospital there had only two thermometers, and the administrator’s best assistant, “thoroughly discouraged”, had committed suicide the day before.

In Moscow, things were better, but even among the reasonably well provided-for members of the Russian Academy of Sciences, 21 had died from disease and malnutrition.

Golder got the impression that Sergei Oldenburg, the Permanent Secretary of the Academy, would soon be joining them. “It is so pitiful and so heart breaking that it completely upsets me,” Golder wrote, recalling his visit for his friend and patron Ephraim D. Adams. “I wish you could meet him for he is one of the most scholarly, cultured and kindly men that I have ever met.” Oldenburg, nearly 60 by then, was bedridden. He was barely able to reach the scraps of toast on his table, let alone chew them. He had spent the past four years trying to support his own family, the orphan children of his brother, his daughter-in-law and his grandchildren, all on about nine million roubles – or five US dollars – a month. The children had bread every day: for the rest of the family, it was a weekly treat.

Sergei Oldenburg was one of the privileged ones: highly-educated, patrician, a personal acquaintance of Vladimir Lenin, and engaged in work vital to the state. Along with other scholars, he had even been recieving a dole, although, as he wrote, “it needed the limitless authority of Lenin and the enormous popularity of Gorky to carry off the issuing of an ‘academic ration’. For this exceptional ration was created before the eyes of the hungry masses who had set themselves the task of destroying all privileges and hierarchies.”

Class resentment, exacerbated by the emergency, was indeed fierce: Frank Golder recalls how one professor called upon the representative of the Crimean government – a young female Communist – to point out that professors in the Crimea were dying of hunger. “What of it?” she had replied. “Let them die!”

Since the October Revolution, Sergei Oldenburg had worked “like a giant” trying to keep up the Academy, trying to find the Academicians something to eat, trying to keep on good terms with the Bolsheviks while striving not to alienate the anti-Bolsheviks. Oldenburg, a world-renowned Orientalist, grandson of a Full General in the Imperial Russian Army, and with a modest amount of blue blood running through his veins, was a liberal nationalist; never a communist. In 1905 he had served in the Russian Provisional Government as Minister of Education. Unlike his political colleagues, however, he had chosen to remain in Russia following the Bolshevik takeover. When Golder, seated at Oldenburg’s bedside in Moscow, talked strenuously about American state recognition, and the good American investment capital might do to save the country, Oldenburg’s mixture of national pride, and his precoccuption with the redemptive powers of suffering, marked him as a leftover of a bygone age:

“Our salvation can not come from without but must come from within and we, as a government and to some extent as a nation have not yet confessed and repented our sins… Let us recover slowly, let us suffer some more the cruel pangs of hunger because it is the only way to get well and strong… All the suffering, all the misery we have endured and are enduring is teaching us Russians to think clearly and that is a great step in the line of progress.”

Oldenburg was certainly a clear thinker: a scholar of Buddhism who welcomed and enjoyed the company of the growing number of Academicians who were natural scientists. The Academy itself was old, founded in 1724 by Peter the Great. It had always been a more reliable friend to the State than the universities, and had enjoyed a privileged position, as a sort of expert arm of the Russian civil service. Oldenburg knew how to convey the Academy’s value to those now in power. He emphasised the practical benefits working with the Academy. His close friend Vladimir Vernadsky had established the Academy’s Commission for the Study of Natural Productive Forces (KEPS) – a key asset in negotiations with the government. Oldenburg asked for money and independence; in return, he could help the State develop greater self-sufficiency in raw materials and manufactures, and even help Lenin with his over-ambitious plan for the rapid electrification of Russia.
It was never an easy compromise, but Lenin, for his part, understood how important the Academy was to the Russia’s survival. (When, in 1922, a Proletkult bigwig wrote a Pravda article hostile to the Academy, Lenin, unimpressed, scrawled in the margin: “And what percentage of [his] loyal proliterians know how to build locomotives?”) So Sergei Oldenburg survived the famine, the flood that inundated his apartment in 1924, and even the attentions of “that black cloud from Moscow”, the astronomer Vartan Ter-Oganezov, an ideologue whose ambitions to remake science in the image of Bolshevism earn him a chapter later in this account. In this chapter, we will see how Oldenburg, with astonishing political dexterity, shaped the future of the world’s largest scientific institution: a sprawling organisation that fed and clothed almost all the people whose lives and and careers are described in this book.

From the beginning, Russian scientists had reservations about communist ideology. Until the “Great Break” and Cultural Revolution of 1929 there was not one member of the Academy of Sciences who was also a member of the Communist Party. But few Academicians could resist the allure of Sergei Oldenburg’s vision of the the Academy’s future: a scientistic programme of modernisation that offered many influential positions to the scientists and engineers willing to work with the communist government. The new Academy grew vast, comprising hundreds of research institutes spread across the USSR. Its central control structure appealed to Lenin’s notorious successor Joseph Stalin. But it appealed just as much to Academicians of Oldenburg’s stripe and generation: men who, in Tsarist times. had argued for nothing else.
Class resentment, wielded as a weapon by Joseph Stalin, eventually destroyed the arrangements Oldenburg spent so many years maintaining. Oldenburg was dismissed from his post during the “Great Break” of 1929. His legacy lived on, nonetheless: a collosal working institution, often troubled, often compromised, but recognised the world over as a pillar of world science.

Unknown Quantity: a Real and Imagined History of Algebra by John Derbyshire

Unknown Quantity: a Real and Imagined History of Algebra by John Derbyshire
reviewed for the Telegraph,  17 May 2007

In 1572, the civil engineer Rafael Bombelli published a book of algebra, which, he said, would enable a novice to master the subject. It became a classic of mathematical literature. Four centuries later, John Derbyshire has written another complete account. It is not, and does not try to be, a classic. Derbyshire’s task is harder than Bombelli’s. A lot has happened to algebra in the intervening years, and so our expectations of the author – and his expectations of his readers – cannot be quite as demanding. Nothing will be mastered by a casual reading of Unknown Quantity, but much will be glimpsed of this alien, counter-intuitive, yet extremely versatile technique.

Derbyshire is a virtuoso at simplifying mathematics; he is best known for Prime Obsession (2003), an account of the Riemann hypothesis that very nearly avoided mentioning calculus. But if Prime Obsession was written in the genre of mathematical micro-histories established by Simon Singh’s Fermat’s Last Theorem, Derbyshire’s new work is more ambitious, more rigorous and less cute.

It embraces a history as long as the written record and its stories stand or fall to the degree that they contribute to a picture of the discipline. Gone are Prime Obsession’s optional maths chapters; in Unknown Quantity, six “maths primers” preface key events in the narrative. The reader gains a sketchy understanding of an abstract territory, then reads about its discovery. This is ugly but effective, much like the book itself, whose overall tone is reminiscent of Melvyn Bragg’s Radio 4 programme In Our Time: rushed, likeable and impossibly ambitious.

A history of mathematicians as well as mathematics, Unknown Quantity, like all books of its kind, labours under the shadow of E T Bell, whose Men of Mathematics (1937) set a high bar for readability. How can one compete with a description of 19th-century expansions of Abel’s Theorem as “a Gothic cathedral smothered in Irish lace, Italian confetti and French pastry”?

If subsequent historians are not quite left to mopping-up operations, it often reads as though they are. In Unknown Quantity, you can almost feel the author’s frustration as he works counter to his writerly instinct (he is also a novelist), applying the latest thinking to his biography of the 19th-century algebraist Évariste Galois – and draining much colour from Bell’s original.

Derbyshire makes amends, however, with a few flourishes of his own. Also, he places himself in his own account – a cultured, sardonic, sometimes self-deprecating researcher. This is not a chatty book, thank goodness, but it does possess a winning personality.

Sometimes, personality is all there is. The history of algebra is one of stops and starts. Derbyshire declares that for 269 years (during the 13th, 14th and early 15th centuries) little happened. Algebra is the language of abstraction, an unnatural way of thinking: “The wonder, to borrow a trope from Dr Johnson, is not that it took us so long to learn how to do this stuff; the wonder is that we can do it at all.”

The reason for algebra’s complex notation is that, in Leibniz’s phrase, it “relieves the imagination”, allowing us to handle abstract concepts by manipulating symbols. The idea that it might be applicable to things other than numbers – such as sets, and propositions in logic – dawned with tantalising slowness. By far the greater part of Derbyshire’s book tells this tale: how mathematicians learned to let go of number, and trust the terrifying fecundity of their notation.

Then, as we enter the 20th century, and algebra’s union with geometry, something odd happens: the mathematics gets harder to do but easier to imagine. Maths, of the basic sort, is a lousy subject to learn. Advanced mathematics is rich enough to sustain metaphor, so it is in some ways simpler to grasp.

Derbyshire’s parting vision of contemporary algebra – conveyed through easy visual analogies, judged by its applicability to physics, realised in glib computer graphics – is almost a let-down. The epic is over. The branches of mathematics have so interpenetrated each other, it seems unlikely that algebra, as an independent discipline, will survive.

This is not a prospect Derbyshire savours, which lends his book a mordant note. This is more than an engaging history; it records an entire, perhaps endangered, way of thinking.