With the stroke of a pen the Jesuit fathers banned the doctrine of infinitesimals, announcing that it could never be taught or even mentioned. The concept was deemed dangerous and subversive, a threat to the belief that the world was an orderly place, governed by a strict and unchanging set of rules. If infinitesimals were ever accepted, the Jesuits feared, the entire world would be plunged into chaos.In Infinitesimal, the award-winning historian Amir Alexander exposes the deep-seated reasons behind the rulings of the Jesuits and shows how the doctrine persisted, becoming the foundation of calculus and much of modern mathematics and technology. Indeed, not everyone agreed with the Jesuits. Philosophers, scientists, and mathematicians across Europe embraced infinitesimals as the key to scientific progress, freedom of thought, and a more tolerant society. As Alexander reveals, it wasn't long before the two camps set off on a war that pitted Europe's forces of hierarchy and order against those of pluralism and change.The story takes us from the bloody battlefields of Europe's religious wars and the English Civil War and into the lives of the greatest mathematicians and philosophers of the day, including Galileo and Isaac Newton, Cardinal Bellarmine and Thomas Hobbes, and Christopher Clavius and John Wallis. In Italy, the defeat of the infinitely small signaled an end to that land's reign as the cultural heart of Europe, and in England, the triumph of infinitesimals helped launch the island nation on a course that would make it the world's first modern state.From the imperial cities of Germany to the green hills of Surrey, from the papal palace in Rome to the halls of the Royal Society of London, Alexander demonstrates how a disagreement over a mathematical concept became a contest over the heavens and the earth. The legitimacy of popes and kings, as well as our beliefs in human liberty and progressive science, were at stake-the soul of the modern world hinged on the infinitesimal.

    . . full of intellectual treats and tricks, of whimsy and deep scientific philosophy. It is highbrow entertainment at its best, a teasing challenge to all who aspire to think about the universe." — New York Herald TribuneOne of the world's foremost nuclear physicists (celebrated for his theory of radioactive decay, among other accomplishments), George Gamow possessed the unique ability of making the world of science accessible to the general reader.He brings that ability to bear in this delightful expedition through the problems, pleasures, and puzzles of modern science. Among the topics scrutinized with the author's celebrated good humor and pedagogical prowess are the macrocosm and the microcosm, theory of numbers, relativity of space and time, entropy, genes, atomic structure, nuclear fission, and the origin of the solar system.In the pages of this book readers grapple with such crucial matters as whether it is possible to bend space, why a rocket shrinks, the "end of the world problem," excursions into the fourth dimension, and a host of other tantalizing topics for the scientifically curious. Brimming with amusing anecdotes and provocative problems, One Two Three . . . Infinity also includes over 120 delightful pen-and-ink illustrations by the author, adding another dimension of good-natured charm to these wide-ranging explorations.Whatever your level of scientific expertise, chances are you'll derive a great deal of pleasure, stimulation, and information from this unusual and imaginative book. It belongs in the library of anyone curious about the wonders of the scientific universe. "In One Two Three . . . Infinity, as in his other books, George Gamow succeeds where others fail because of his remarkable ability to combine technical accuracy, choice of material, dignity of expression, and readability." — Saturday Review of Literature

    We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous. Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greece and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS. As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.

    Augustine equate nothingness with the Devil? What tortuous means did 17th-century scientists employ in their attempts to create a vacuum? And why do contemporary quantum physicists believe that the void is actually seething with subatomic activity? You’ll find the answers in this dizzyingly erudite and elegantly explained book by the English cosmologist John D. Barrow.Ranging through mathematics, theology, philosophy, literature, particle physics, and cosmology, The Book of Nothing explores the enduring hold that vacuity has exercised on the human imagination. Combining high-wire speculation with a wealth of reference that takes in Freddy Mercury and Shakespeare alongside Isaac Newton, Albert Einstein, and Stephen Hawking, the result is a fascinating excursion to the vanishing point of our knowledge.

    Having just completed his masterpiece, The General Theory of Relativity—which provided a brand-new theory of gravity and promised a new perspective on the cosmos as a whole—he set out at once to share his excitement with as wide a public as possible in this popular and accessible book.Here published for the first time as a Penguin Classic, this edition of Relativity features a new introduction by bestselling science author Nigel Calder.

    In Turing’s Cathedral, George Dyson focuses on a small group of men and women, led by John von Neumann at the Institute for Advanced Study in Princeton, New Jersey, who built one of the first computers to realize Alan Turing’s vision of a Universal Machine. Their work would break the distinction between numbers that mean things and numbers that do things—and our universe would never be the same. Using five kilobytes of memory (the amount allocated to displaying the cursor on a computer desktop of today), they achieved unprecedented success in both weather prediction and nuclear weapons design, while tackling, in their spare time, problems ranging from the evolution of viruses to the evolution of stars. Dyson’s account, both historic and prophetic, sheds important new light on how the digital universe exploded in the aftermath of World War II. The proliferation of both codes and machines was paralleled by two historic developments: the decoding of self-replicating sequences in biology and the invention of the hydrogen bomb. It’s no coincidence that the most destructive and the most constructive of human inventions appeared at exactly the same time.  How did code take over the world? In retracing how Alan Turing’s one-dimensional model became John von Neumann’s two-dimensional implementation, Turing’s Cathedral offers a series of provocative suggestions as to where the digital universe, now fully three-dimensional, may be heading next.

    In his delightfully down-to-earth style, he explores & explains a mind-boggling future of intelligent robots, extraterrestrial life & its consquences, & other provocative, fascinating quandries of the future we want to see today.

    This discovery dramatically reshaped how humans understood their place in the cosmos, and once and for all laid to rest the idea that the Milky Way galaxy was alone in the universe. Six years later, continuing research by Hubble and others forced Albert Einstein to renounce his own cosmic model and finally accept the astonishing fact that the universe was not immobile but instead expanding. The fascinating story of these interwoven discoveries includes battles of will, clever insights, and wrong turns made by the early investigators in this great twentieth-century pursuit. It is a story of science in the making that shows how these discoveries were not the work of a lone genius but the combined efforts of many talented scientists and researchers toiling away behind the scenes. The intriguing characters include Henrietta Leavitt, who discovered the means to measure the vast dimensions of the cosmos . . . Vesto Slipher, the first and unheralded discoverer of the universe’s expansion . . . Georges Lemaître, the Jesuit priest who correctly interpreted Einstein’s theories in relation to the universe . . . Milton Humason, who, with only an eighth-grade education, became a world-renowned expert on galaxy motions . . . and Harlow Shapley, Hubble’s nemesis, whose flawed vision of the universe delayed the discovery of its true nature and startling size for more than a decade.Here is a watershed moment in the history of astronomy, brought about by the exceptional combination of human curiosity, intelligence, and enterprise, and vividly told by acclaimed science writer Marcia Bartusiak.

    In Why Beauty Is Truth, world-famous mathematician Ian Stewart narrates the history of the emergence of this remarkable area of study. Stewart introduces us to such characters as the Renaissance Italian genius, rogue, scholar, and gambler Girolamo Cardano, who stole the modern method of solving cubic equations and published it in the first important book on algebra, and the young revolutionary Evariste Galois, who refashioned the whole of mathematics and founded the field of group theory only to die in a pointless duel over a woman before his work was published. Stewart also explores the strange numerology of real mathematics, in which particular numbers have unique and unpredictable properties related to symmetry. He shows how Wilhelm Killing discovered “Lie groups” with 14, 52, 78, 133, and 248 dimensions-groups whose very existence is a profound puzzle. Finally, Stewart describes the world beyond superstrings: the “octonionic” symmetries that may explain the very existence of the universe.

    This book can be cut, drawn in, folded into shapes and will even take you to the fourth dimension. So join stand-up mathematician Matt Parker on a journey through narcissistic numbers, optimal dating algorithms, at least two different kinds of infinity and more.

    . . a book that enriches our understanding of how the struggle to think new thoughts is shared across time and space and people.”—The Sunday Telegraph (London)Christmas, 1859. Just one month after the publication of On the Origin of Species, Charles Darwin received an unsettling letter. He had expected criticism; in fact, letters were arriving daily, most expressing outrage and accusations of heresy. But this letter was different. It accused him of failing to acknowledge his predecessors, of taking credit for a theory that had already been discovered by others. Darwin realized that he had made an error in omitting from Origin of Species any mention of his intellectual forebears. Yet when he tried to trace all of the natural philosophers who had laid the groundwork for his theory, he found that history had already forgotten many of them.Darwin’s Ghosts tells the story of the collective discovery of evolution, from Aristotle, walking the shores of Lesbos with his pupils, to Al-Jahiz, an Arab writer in the first century, from Leonardo da Vinci, searching for fossils in the mine shafts of the Tuscan hills, to Denis Diderot in Paris, exploring the origins of species while under the surveillance of the secret police, and the brilliant naturalists of the Jardin de Plantes, finding evidence for evolutionary change in the natural history collections stolen during the Napoleonic wars. Evolution was not discovered single-handedly, Rebecca Stott argues, contrary to what has become standard lore, but is an idea that emerged over many centuries, advanced by daring individuals across the globe who had the imagination to speculate on nature’s extraordinary ways, and who had the courage to articulate such speculations at a time when to do so was often considered heresy.With each chapter focusing on an early evolutionary thinker, Darwin’s Ghosts is a fascinating account of a diverse group of individuals who, despite the very real dangers of challenging a system in which everything was presumed to have been created perfectly by God, felt compelled to understand where we came from. Ultimately, Stott demonstrates, ideas—including evolution itself—evolve just as animals and plants do, by intermingling, toppling weaker notions, and developing over stretches of time. Darwin’s Ghosts presents a groundbreaking new theory of an idea that has changed our very understanding of who we are.

    Admittedly real life wasn’t like that. But only, they argued, because we didn’t have enough data to be certain.Then the cracks began to appear. It proved impossible to predict exactly how three planets orbiting each other would move. Meteorologists discovered that the weather was truly chaotic – so dependent on small variations that it could never be predicted for more than a few days out. And the final nail in the coffin was quantum theory, showing that everything in the universe has probability at its heart.That gives human beings a problem. We understand the world through patterns. Randomness and probability will always be alien to us. But it’s time to plunge into this fascinating, shadowy world, because randomness crops up everywhere. Probability and statistics are the only way to get a grip on nature’s workings. They may even seal the fate of free will and predict how the universe will end.Forget Newton’s clockwork universe. Welcome to Dice World.

    Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.Brought up to date with a new chapter by Ian Stewart, What is Mathematics? Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.Formal mathematics is like spelling and grammar - a matter of the correct application of local rules. Meaningful mathematics is like journalism - it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature - it opens a window onto the world of mathematics for anyone interested to view.

    Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world.

    As we enter the year 2000, zero is once again making its presence felt. Nothing itself, it makes possible a myriad of calculations. Indeed, without zero mathematicsas we know it would not exist. And without mathematics our understanding of the universe would be vastly impoverished. But where did this nothing, this hollow circle, come from? Who created it? And what, exactly, does it mean? Robert Kaplan's The Nothing That Is: A Natural History of Zero begins as a mystery story, taking us back to Sumerian times, and then to Greece and India, piecing together the way the idea of a symbol for nothing evolved. Kaplan shows us just how handicapped our ancestors were in trying to figurelarge sums without the aid of the zero. (Try multiplying CLXIV by XXIV). Remarkably, even the Greeks, mathematically brilliant as they were, didn't have a zero--or did they? We follow the trail to the East where, a millennium or two ago, Indian mathematicians took another crucial step. By treatingzero for the first time like any other number, instead of a unique symbol, they allowed huge new leaps forward in computation, and also in our understanding of how mathematics itself works. In the Middle Ages, this mathematical knowledge swept across western Europe via Arab traders. At first it was called dangerous Saracen magic and considered the Devil's work, but it wasn't long before merchants and bankers saw how handy this magic was, and used it to develop tools likedouble-entry bookkeeping. Zero quickly became an essential part of increasingly sophisticated equations, and with the invention of calculus, one could say it was a linchpin of the scientific revolution. And now even deeper layers of this thing that is nothing are coming to light: our computers speakonly in zeros and ones, and modern mathematics shows that zero alone can be made to generate everything.Robert Kaplan serves up all this history with immense zest and humor; his writing is full of anecdotes and asides, and quotations from Shakespeare to Wallace Stevens extend the book's context far beyond the scope of scientific specialists. For Kaplan, the history of zero is a lens for looking notonly into the evolution of mathematics but into very nature of human thought. He points out how the history of mathematics is a process of recursive abstraction: how once a symbol is created to represent an idea, that symbol itself gives rise to new operations that in turn lead to new ideas. Thebeauty of mathematics is that even though we invent it, we seem to be discovering something that already exists.The joy of that discovery shines from Kaplan's pages, as he ranges from Archimedes to Einstein, making fascinating connections between mathematical insights from every age and culture. A tour de force of science history, The Nothing That Is takes us through the hollow circle that leads to infinity.