Leonard Euler

Sophie once again steps outside of her comfort zone to examine the life and myriad contributions of famed Swiss mathematician Leonard Euler.  Euler not only introduced much of our modern mathematical notation but also defined the value of essential constants and published prolifically.

The parallels, including the multi-disciplinary expertise, of Sophie's choice as modern-era contrasting figure - an individual whose contributions to computer science, game theory and numerical analysis are both fascinating and largely unheralded - are profound.

This is a work of fiction. Any resemblance to actual persons, living or dead, or actual events is purely coincidental.

Music by Gregor Quendel from Pixabay 

https://www.buzzsprout.com/2599997/episodes/18768959

Transcript

SPEAKER_00 0:00

Leonhard Euler and the Unreasonable Productivity of Genius, a podcast by Sophie Manners. Hello and welcome back. I'm Sophie Manners, Professor of European History at Queen's College, Cambridge, author of biographies of people who changed the world, wearer of a scarf that a mathematician colleague described this morning as topologically interesting, which I suspect was a compliment of some kind, and today you'll guide through a life that requires me to do something I have not done in any previous episode of this series. I need to ask you to take several things on faith, not religious faith, not the faith of Cromwell or Joan of Arc, but the specific faith that a non-specialist extends toward a specialist, when the specialist says, Trust me, this is extraordinary, because today's subject is a mathematician, the greatest mathematician in the history of the discipline, by the assessment of most people qualified to make that assessment, and by a margin that is not especially close. And the nature of his greatness is such that explaining it fully requires mathematical knowledge that I do not possess, and that most of my listeners do not possess, and that the podcast format cannot adequately convey in any case. What I can do, what I intend to do, is explain why it matters. Why a Swiss mathematician who spent most of his adult life in St. Petersburg and Berlin, who went blind at fifty nine and kept working, who had thirteen children, and apparently found them a pleasant background to his calculations, who wrote so much that the Academy publishing his collected works has been at it since nineteen eleven and is not yet finished. Why this man is not merely a figure in the history of mathematics, but one of the most consequential human beings who ever lived in ways that are present in your life today, whether or not you have ever thought about mathematics, and whether or not you have any intention of starting. He is not a tragic figure, he is not a controversial figure, he is not a martyr or a conqueror, or a revolutionary, and he spent no time in prison and did not order anyone executed and did not dissolve a parliament or shoot anyone at waxaws. He is something rarer and in a particular way more extraordinary than any of those things. He is a man who sat down and thought for sixty years and whose thinking reshaped the foundations of human knowledge. I find this, I will tell you plainly, more impressive than most things I have encountered in this series, and I have spent the last year discussing Alexander the Great, Leonhard Euler, born april fifteenth, seventeen oh seven, in Basel, Switzerland, died september eighteenth, seventeen eighty three, in St. Petersburg, Russia, seventy six years old. In the interval, he produced somewhere between eight hundred and nine hundred published mathematical works, papers, books, memoirs, letters, a quantity so prodigious that mathematicians still use the shorthand the Euler Opera Omnia for the collected edition of his works, which runs to more than eighty volumes, and which the editors, who began their labour in nineteen eleven, have not yet completed. He contributed foundational work to calculus, number theory, geometry, graph theory, topology, mechanics, optics, astronomy, fluid dynamics, and music theory. He introduced the notation that every person who has ever taken a mathematics class uses without thinking f of x for a function, e for the base of natural logarithms, i for the square root of negative one, the use of the Greek letter pi for the ratio of a circle's circumference to its diameter, the capital Sigma for summation. He did not discover all of these. He standardized and popularized them, which is in many respects a greater contribution to the actual practice of mathematics. Since a tool that everyone uses in the same way is infinitely more powerful than a tool each person reinvents for themselves, he also did all of this for the last seventeen years of his life, entirely blind. Let us begin in Basel, which is a city on the Rhine in what is now northwestern Switzerland, that has been producing remarkable people for centuries, and that takes this with the particular Swiss equanimity that suggests it finds the whole business mildly tiring, but professionally useful. Euler's father, Paul, was a Protestant minister who had studied mathematics at the University of Basel under Jakob Bernoulli, one of the most distinguished mathematicians in Europe, and who passed his son both the theological seriousness of the Calvinist tradition and a grounding in elementary mathematics that gave Leonhard a foundation. Most children of his era could not have imagined Paul Euler intended his son for the ministry. The son had other ideas, which became clear with a speed that the University of Basel found impressive, and Paul Euler found eventually persuasive. At thirteen, Euler enrolled at the University of Basel. This is not as alarming as it sounds. University entry ages in the early eighteenth century were lower than we now expect, but his subsequent progress was alarming by any standard. He completed the standard arts curriculum by fifteen. By sixteen he had attracted the serious attention of Johann Benoulli, son of Jakob, and himself one of the preeminent mathematicians of the early eighteenth century. Johann Benouly agreed to give Euler private lessons on Saturday afternoons, not because this was a standard arrangement, but because Euler's ability was such that Bernoulli concluded the university curriculum was simply not adequate to what the boy needed, and that the most useful thing Bernoulli could do was point him at difficult problems and get out of the way. He was seventeen when he completed his master's degree with a dissertation comparing the philosophies of Descartes and Newton. He was nineteen when he wrote his first significant mathematical paper. By twenty he was corresponding with the leading mathematicians of Europe as a near equal. Paul Euler, confronted with evidence of this scale, revised his view about the ministry. Leonhard became a mathematician. In seventeen twenty seven, at twenty years old, Euler moved to St. Petersburg, where Peter the Great's Academy of Sciences had been established, and where his friend Daniel Benoulli, Johann's son, was already working. Russia had, under Peter and his successors, made a deliberate and expensive effort to import European intellectual talent, offering salaries and conditions that the cramped and competitive universities of Western Europe could not match. The Academy of Sciences was a genuine centre of mathematical and scientific work, and for Euler, arriving as a very young man of exceptional gifts, it provided exactly what he needed resources, colleagues, time, and the institutional security to simply work. He was initially appointed to the physiology division, which must have been a brief administrative comedy for anyone who knew him, and was transferred to mathematics within months, which is the only sensible resolution. In 1733, at twenty-six, he was appointed to the chair of mathematics, succeeding Daniel Benoli, who had returned to Basel. He was, at twenty-six, the senior mathematician at one of the leading scientific institutions in the world. In seventeen thirty four he married Katharina Gzell, the daughter of a Swiss painter working in St. Petersburg. The marriage was, by all accounts, a genuinely happy one. They had thirteen children, of whom five survived to adulthood, the child mortality rate of the period being what it was, which is one of those facts about early modern European life that always brings me up short, however many times I encounter it. Euler was, by the accounts of those who knew him, a man of genuine warmth and domestic contentment. He famously did much of his mathematical work, with children playing around his feet, apparently finding them no distraction whatsoever, which tells you either about his powers of concentration or about his relationship to background noise, and I inclined toward both. In 1735 he suffered a severe fever that left him blind in his right eye. He responded to this with characteristic equanimity, reportedly joking that he would have less distraction, a remark that is either genuinely phlegmatic or the kind of statement a deeply focused person makes when they have long since ceased to notice the external world in any particularly acute way. He was twenty eight years old. He continued working at the same pace. The Seven Bridges of Königsberg problem, which Euler solved in 1736, is perhaps the best available illustration of how his mathematical mind worked, and it is worth explaining in some detail because it is both accessible and genuinely beautiful. The city of Königsberg, now Kaliningrad, in Russia, was built on a river that divided it into four distinct land areas, connected by seven bridges. The residents of Königsberg had amused themselves, apparently for some time, with the question of whether it was possible to walk through the city, crossing each of the seven bridges exactly once, without crossing any bridge twice. Nobody had found a route. Nobody had proved one didn't exist. Euler's approach was the thing that made the solution extraordinary. He did not try to find the route or to try all possible routes systematically, or to engage with the specific geography of Königsberg at all in the way the problem as stated seemed to require. He abstracted it, he recognised that the specific details, the streets, the buildings, the exact positions of the bridges, were irrelevant to the mathematical question. What mattered was only the connectivity, how many bridges connected to each land area, and what the structure of those connections was. He replaced the map with what we would now call a graph, a set of points representing land areas, connected by lines, representing bridges, and he proved from the properties of that abstract graph that no route of the required type could exist, because each land area was connected to an odd number of bridges, and any route that crosses each bridge exactly once requires all intermediate points to have an even number of connections. This may sound simple, it is not simple, or rather, it is simple in the way that all genuinely profound insights are simple in retrospect. Which is to say, it was not obvious until Euler made it obvious, and the making of it obvious required the specific leap of seeing that the right way to think about a problem of geography was as a problem of abstract structure. The field of mathematics that Euler invented in the process of solving this puzzle is graph theory, and graph theory is now the foundational mathematics of computer science, internet routing, social network analysis, and almost every other domain of modern technology that involves understanding how things connect. When Google maps a route, when Facebook suggests a friend, when a packet of data finds its way across the internet. The underlying mathematics is graph theory, and graph theory begins with Euler and Seven Bridges and a city that is now in Russia. He left St. Petersburg in 1741, invited to Berlin by Frederick the Great of Prussia, who was engaged in the competitive European business of assembling the finest intellectual talent available at his court, and was not the sort of man to leave the world's greatest mathematician in St. Petersburg when he could have him in Berlin. Euler spent twenty five years at the Berlin Academy of Sciences, years of extraordinary productivity, during which he published more than 380 works and contributed to an almost bewildering range of fields. He produced the introductio in Annalesin Infinitorum in 1748, which is the book that effectively created the modern subject of mathematical analysis, defining functions, establishing the properties of exponential and trigonometric functions, and introducing the notation that mathematics still uses. He wrote textbooks on differential calculus and integral calculus that remained standard references for a century. He worked on optics, on ship design, on music theory, on the mathematics of lotteries, on the motion of the planets, on the construction of canals. He also found time, during the Berlin years, to correspond with a fifteen-year-old German princess, Friederike Charlotte of Brandenburg Schwedt, whose guardians had asked Euler to provide her with a scientific education by letter. The resulting correspondence, eventually published as The Letters to a German Princess, became one of the most widely read scientific popularizations of the eighteenth century, translated into eight languages and running to multiple editions. Euler, it turned out, was not only a mathematician of extraordinary power, but a writer of unusual clarity, capable of explaining difficult ideas to non specialists in language that was precise and accessible, and sometimes in the better passages, genuinely beautiful. The letters are still readable. I have read several of them. They are the work of a man who loved his subject and believed, with the specific generosity of genuine expertise, that everyone deserved access to it. His relationship with Frederick the Great was more complicated than his relationship with the princess. Frederick was an admirer of French intellectual culture, and specifically of Voltaire, and he found Euler, who was religious, domestic, plain spoken, and almost entirely uninterested in the social performance of the Prussian court. Somewhat puzzling, a famous story, which may be apocryphal, but which has the quality of truth, holds that Voltaire once asked Euler why he spoke so little at court, and Euler replied Because I come from a country where one speaks only when one has something to say. This is either apocryphal or the most Eulerian thing that Euler ever said, and both possibilities are equally satisfying. He returned to St. Petersburg in seventeen sixty six, at fifty nine, invited by Catherine the Great, who was conducting her own version of the competitive European business of assembling intellectual talent. Shortly after his return, he lost the sight in his remaining eye, a cataract, and a subsequent failed operation to correct it. He was completely blind. He did not stop working. I want to pause here because I think this fact requires more attention than it typically receives. Leonhard Euler, in the last seventeen years of his life, completely blind, produced approximately half of his total lifetime output. He did this by dictating to assistants and scribes, by performing complex calculations entirely in his head, by drawing on a mathematical memory and a mathematical imagination of a quality that defies easy analogy. He had, by this stage in his life, so thoroughly internalized the structures of mathematics that he did not require visual access to notation to do mathematics. The notation was, by this point, so fully internal that the absence of external representation was not an obstacle. He simply thought. The thinking happened to be the most productive mathematical thinking in history. A house fire in seventeen seventy one destroyed much of his property, and Euler himself was rescued from the burning building by a craftsman from Basel. Within weeks he had resumed work. He dictated a paper on a complex mathematical problem to his son during the rescue. Whether this is literally accurate or the kind of biographical legend that gathers around very large figures, I cannot say with certainty, but it is consistent with everything we know about him, and the metaphor it carries the fire, the rescue, the continuation of work is Eulerian in its totality. Catherina died in 1773 after nearly forty years of marriage. Euler grieved by the accounts of those close to him with genuine depth. He remarried in 1776. Salome Abigail Gizelle, Catherina's half sister, which is the kind of biographical detail that the eighteenth century took with more equanimity than the twenty first century typically manages, and which I mention primarily because it suggests a man who valued domestic stability and companionship not as a background condition to work, but as something he actively sought and appreciated. He died on september eighteenth, seventeen eighty three, in St. Petersburg. He had spent the day working, calculating the orbit of Uranus, recently discovered by William Herschel, the sort of new problem that could have occupied a lesser mathematician for months, and that Euler apparently treated as an interesting afternoon's work. He suffered a stroke in the water. Dinner, was supported by his assistant, said something that his assistant's account renders as I am dying, and died. He was seventy six years old. The Marquis de Condorcet, in his eulogy to Euler for the French Academy, wrote He ceased to calculate and to live. It is one of the better obituary sentences in the history of mathematics. Now Euler's mathematical legacy I promised you that this would matter to your life, and I intend to honour that promise, but doing so requires me to explain briefly what he actually did, because the scale of it is not intuitive, and the importance of it is not obvious unless you trace the lines forward. The notation I mentioned earlier, F of X EI Pi, sigma is not a trivial contribution. Mathematics is, in its practice, a language, and the language has to be learned before the ideas can be communicated. Before Euler standardized these symbols, the same mathematical objects were referred to by different names and different symbols in different traditions and different countries, and the work of mathematical communication required constant translation. After Eler, the language was in its essentials unified, and mathematical ideas could travel from one country and one tradition to another without the friction of notational incompatibility. This is roughly equivalent in its effect on the practice of mathematics to what the printing press did to the production of books. The ideas existed before Euler. The universally shared language for expressing them did not. The Euler identity a to the power of i times pi plus one equals zero was voted in a nineteen ninety poll of mathematicians, the most beautiful equation in mathematics. It brings together five of the most fundamental numbers in all of mathematics a the base of natural logarithms, i the imaginary unit, the square root of negative one, pi, the ratio of a circle's circumference to its diameter, one, the multiplicative identity, and zero, the additive identity. These five numbers arise from entirely different areas of mathematics, exponential growth, imaginary numbers, geometry, arithmetic, and they are connected by this single equation in a way that, when mathematicians first encounter it, tends to produce either speechlessness or the specific quality of intellectual pleasure that most people associate with great art rather than with calculation. A Stanford professor, Keith Devlin, described it as something that reaches down into the very depths of existence. I am a historian, not a mathematician, and I cannot verify that claim from the inside. But I can tell you that the number of people who've been moved, genuinely moved, by a mathematical equation is not large, and that an equation capable of producing that response is doing something important. Graph theory, which began with the Koenigsberg Bridges, now underlies every computer network, every internet protocol, every algorithm that routes data across the globe, every social network analysis that attempts to understand how information and influence travel through populations, when epidemiologists model the spread of infectious disease through contact networks, they are using graph theory. When logistics companies optimize delivery routes, they are using graph theory. When search engines rank web pages, by how many other pages link to them, they are using graph theory. The bridges of Koenigsberg have never stopped being crossed. Euler's work on number theory, the properties of prime numbers, the distribution of primes, the relationship between primes and the structure of integers is the foundation on which modern cryptography is built, which means it is the foundation on which secure internet communication, online banking, digital privacy, and the entire infrastructure of the modern digital economy rests. Every time you send an encrypted message, every time you make an online purchase, every time you connect to a website via a secure protocol, you are doing something that is mathematically downstream of Euler. He never knew there would be an internet. He was thinking about prime numbers because prime numbers were interesting. The practical application arrived 200 years later and found his mathematics already waiting. His work on fluid dynamics is the foundation of modern aerodynamics. His work on mechanics is taught in every physics and engineering curriculum. His work on the calculus of variations, finding the curves and surfaces that optimise given properties is essential to physics, engineering, economics, and the mathematical modelling of essentially every complex system. He did all of this I want to return to this point because it genuinely requires returning to, while going progressively blind, while raising a large family, while navigating the political complications of working for courts that were in intermittent conflict with each other, while corresponding with virtually every significant mathematician in Europe, while writing accessible popular science for a German princess, while calculating the orbit of Uranus on the afternoon he died, he published more mathematics than any other mathematician in history. The journal Mathematica dedicated a special issue in his honour to examining what he had done. The conclusion was essentially everything. He had worked in every major area of mathematics. He had made significant contributions to almost all of them. He had founded several new ones. Laplas, himself one of the greatest mathematicians of the generation after Euler, said Read Euler, read Euler. He is the master of us all. Now the tease and this is a tease I have been constructing with particular care, because the parallel for Euler is not primarily about tragedy or politics or contested legacy. It is about something more unusual. The specific quality of a mind so productive, so generative, so naturally at home in abstraction, that its output exceeds what the people around it can fully absorb in real time, and whose consequences continue to ripple outward for centuries. Picture someone whose work is so fundamental to modern technology that it is essentially invisible, present everywhere, acknowledged almost nowhere, built into the infrastructure of the world with such completeness that most of the people who depend on it daily have never heard its author's name. Someone whose ideas were not merely ahead of their time, but operating in a register that required the invention of the time before the ideas could be fully applied. Picture someone who was from an early age, so evidently operating at a level beyond what their immediate environment could challenge, that they required a different kind of intellectual companionship, mentors who recognized not that they needed teaching in the conventional sense, but that they needed problems, difficult problems, problems that had not been solved. Someone whose education was not primarily about receiving knowledge, but about being exposed to the frontier where knowledge ended, because that frontier was where they were most alive. Picture someone who was prodigiously productive in multiple domains simultaneously, not a narrow specialist who mastered one field, but someone who moved across fields with a restlessness that looked from the outside less like ambition and more like genuine comprehensive curiosity, someone for whom the categories of knowledge were not the limits of inquiry, but simply the organization of problems already solved. Picture someone who continued to work through conditions that would have ended the careers of most people, physical challenges, personal losses, the loss of the primary sensory modality through which most people in their field engaged with their work, and who did so not through extraordinary willpower or performed heroism, but through the simple, direct mechanism of not having the kind of relationship to their work that made stopping thinkable. Picture someone whose most important contribution is not any single discovery or invention, but a language, a way of describing the world that made subsequent discovery possible, and who is therefore simultaneously celebrated by specialists and invisible to the general public because the language is so thoroughly absorbed into the practice that its origin has become invisible. Picture someone who was, in their personal life, by all accounts warm, domestic, companionable, and almost aggressively normal, someone whose extraordinary gifts coexisted with what their contemporaries found to be a thoroughly decent, thoroughly ordinary human personality, and who seemed, if anything, mildly surprised that anyone found this combination remarkable. Who is this person? It is John von Neumann. I hear the mathematicians and the computer scientists among you nodding, and I hear the historians among you preparing to look him up, and both reactions are appropriate. John von Neumann was born Neumann Janos Lyos on december twenty eighth, nineteen oh three, in Budapest, Hungary, the son of a wealthy Jewish banker. He was, from an early age, a figure of such obvious and comprehensive mathematical talent that the stories about him have the quality of legend. He had memorized a telephone directory by the age of six, read his way through and apparently retained a forty-four volume history of the world in childhood, and was recognised by his teachers as operating by his early teens at a level that the Budapest school system was genuinely not equipped to address. He received his doctoral degree in mathematics from Budapest at twenty-two while simultaneously completing a degree in chemical engineering in Zurich because his father had insisted on a practical qualification as insurance against the possibility that pure mathematics would not pay, and von Neumann found the dual program unproblematic. The chemical engineering degree is perhaps the most charming detail in his biography. Evidence of parental pragmatism applied to a person for whom pragmatism was essentially irrelevant. He moved to Germany, then, as the Nazi threat became impossible to ignore, to the United States in nineteen thirty three, where he became one of the original faculty members of the newly established Institute for Advanced Study in Princeton, alongside Einstein. He spent the rest of his working life in America, dying of bone cancer in Washington, DC in february nineteen fifty seven, at fifty three. He was, like Euler, cut short, though at fifty three rather than seventy-six, and by cancer rather than stroke, and his death was considerably more agonized and considerably less graceful than Euler's, which I mention not to dwell on it, but because the contrast between the deaths of the two men tells you something about the different textures of their lives. What he did in the roughly three decades of his mature working career maps onto Euler's legacy with a structural precision that I find remarkable. First, the foundational contribution that reshaped an entire field's practice. Euler standardized mathematical notation. Von Neumann in the nineteen forties designed the architecture of the modern digital computer. The von Neumann architecture, a stored program computer in which both instructions and data are held in the same memory, is the design on which virtually every general purpose computer built since nineteen forty five has been based. Your laptop, your smartphone, the server processing your financial transactions, the computer in your car. All of them are at the architectural level, von Neumann machines. He did not invent the computer. Alan Turing's theoretical work preceded his, and several engineering teams were working on practical machines simultaneously. But he provided the architectural framework that made general purpose computing feasible, and the importance of that framework is as difficult to overstate as the importance of Euler's notation. Second, the simultaneous work across multiple domains. Euler worked in mathematics, physics, mechanics, optics, astronomy, and music. Von Neumann made foundational contributions to quantum mechanics, game theory, functional analysis, numerical analysis, cellular automata, and the design of nuclear weapons. The breadth is not accidental. Both men appear to have had minds that found the borders between disciplines artificial, that saw problems rather than fields, and that moved toward whichever problems were interesting, regardless of which academic department nominally owned them. Third, the contribution of a language. Euler gave mathematics its notation. Von Neumann, in collaboration with Oscar Morgenstern, published the theory of games and economic behavior in nineteen forty four, which created game theory, the mathematical framework for analysing strategic decision making that is now foundational to economics, political science, evolutionary biology, computer science, and philosophy. Game theory is a language for understanding conflict and cooperation, and like Euler's notation, it has become so thoroughly absorbed into the practice of these fields that most practitioners no longer think of it as a specific intellectual achievement. It is just how you think about strategic problems. Fourth, the continuation through physical limitation. Euler worked blind for seventeen years, von Neumann worked through the bone cancer that was killing him, continuing to think, to calculate, to meet with collaborators, to attend to the problems that interested him until the cancer made it impossible. He dictated his last work, a series of lectures on the relationship between the brain and the computer that were published posthumously as the computer and the brain from his hospital bed. There is something in the quality of devotion to intellectual work that both men exhibit, that goes beyond professionalism into the specific territory of a person for whom thinking is not what they do, but what they are. Fifth, the invisibility of the fundamental. Euler is celebrated by mathematicians and unknown to most people who use mathematics every day. Von Neumann is celebrated by computer scientists and mathematicians, and unknown to most people who use computers every day. The more foundational a contribution, the more invisible it tends to become. Not because the contribution is unappreciated, but because the appreciation takes the form of building on it so thoroughly that the foundation disappears beneath the structure. Where do the parallels strain? Von Neumann worked on the Manhattan Project and the development of thermonuclear weapons, contributing to technologies whose destructive potential he appears to have regarded with less anxiety than many of his colleagues. He was, in this respect, more comfortable than Euler, with the consequences of his work being deployed in ways that were not purely intellectual. Euler was not asked to work on weapons of mass destruction, which is less a moral choice on his part than a function of the era in which he lived. The comparison is complicated by context that is not equivalent. Von Neumann died at fifty three. Euler died at seventy six. The difference of twenty three years is not trivial. Von Neumann's last decade of potential output was taken from him by cancer at exactly the moment when his interests were converging on questions the relationship between computation and biological intelligence, the mathematics of self reproducing systems, that the subsequent development of computer science and artificial intelligence would have made by the nineteen sixties and nineteen seventies exactly the most interesting questions in the world. We do not know what von Neumann would have done with another twenty years. We know what Euler did with his, and the answer is approximately half his total lifetime output. Von Neumann was also, in some respects, that Euler was not, a political figure. He was a consultant to the US military, involved in nuclear weapons development, present at the beginning of the Cold War's technological dimension, in ways that gave his work an entanglement with power and violence that Euler's never had. Euler calculated the orbits of planets and the properties of prime numbers. Von Neumann helped calculate the yield of nuclear explosions. Both were doing what their talents made possible. The political and moral dimensions are not equivalent. The verdict Euler. And I say this not merely because he is the historical subject, but because in the specific terms that I have been applying in this series, the completeness of the arc, the scale of the consequence, the quality of the legacy. Euler is, of all the subjects in this series, the one whose output has the most direct, measurable, and continuous presence in the world as it currently exists. He cannot be diminished by a darker dimension. He has no drogada, he has um no wax horse, he has no Ireland or Vietnam or slave trade advocacy. He sat in a room in St. Petersburg or in Berlin and thought, and what he thought is in every computer. And every network and every encrypted communication and every aerodynamic surface and every prime number and every piece of mathematical notation that has been written since. His blind years were more productive than most mathematicians' entire careers. His lightest published work would constitute a solid professional achievement for a contemporary academic. His greatest work is the foundation of a significant portion of modern technological civilization. He was also apparently very good with children and perfectly happy to have them running around while he worked. I find this detail disproportionately endearing, and I have been in enough seminars at Cambridge to know that the ability to think clearly in conditions of domestic noise is genuinely rare and genuinely valuable. I grew up in Kingston upon Thames. We are not, as I have occasionally mentioned, a town associated with the production of mathematical giants. We are associated with the river and with Scones and with a very pleasant Saturday market, but the router that connects my university office to the internet operates on mathematics that Euler founded. The encryption that protects this podcast from interception uses number theory that Euler developed. The algorithms that recommend it to you use graph theory that Euler invented. He is, in the most direct possible sense, present in the fabric of the world I inhabit, and the world you inhabit, and every world that any person currently alive inhabits, whether or not they know his name, he deserves to be known. He deserves to be celebrated. He deserves and this is not a thing I say about every figure in this series, not just admiration, but something closer to gratitude. Leonhard Euler, born Basel, seventeen oh seven, died St. Petersburg, seventeen eighty three. Mathematician, the most productive scientist in history, the man who gave mathematics its language and then used that language to describe the structure of the world in terms that the world is still learning to fully apply. He ceased to calculate and to live. The calculation has not yet stopped. I'm Sophie Manners. Thank you for being here. If you would like to understand Joyler better than this episode has permitted, and you should want to, because he deserves it, William Dunham's Euler, The Master of Us All, is the best general introduction, and it is written with a warmth and clarity that Euler himself would have approved. Do read it. And if you look at a piece of mathematics tonight, any mathematics, and you see F of X or E or I or the Greek letter pi, that is Euler. He is right there. He has always been right there. Good night.