The triple deck structure shows how a perturbation to the thin layer of a fluid immediately adjacent (boundary layer) to an obstacle such as an aeroplane wing develops as an interaction between the pressure and a displacement of the boundary layer. The theory was a major advance in fluid mechanics at the end of the twentieth century. This book provides a graduate level description of some history of boundary layer flow and the application of the triple deck in fluid mechanics.

Drawing primarily from historical examples, this book explains the tremendous role that numbers and, in particular, mathematics play in all aspects of our civilization and culture. The lively style and illustrative examples will engage the reader who wants to understand the many ways in which mathematics enables science, technology, art, music, politics, and rational foundations of human thought. Each chapter focuses on the influence of mathematics in a specific field and on a specific historical figure, such as "Pythagoras: Numbers and Symbol"; "Bach: Numbers and Music"; "Descartes: Numbers and Space."

Mathematics has long suffered in the public eye through portrayals of mathematicians as socially inept geniuses devoted to an arcane discipline. In this book, Philip J. Davis addresses this image through a question-and-answer dialogue that lays to rest many of the misnomers and misunderstandings of mathematical study. He answers these questions and more: What is Mathematics? Why is mathematics difficult, and why do I spontaneously react negatively when I hear the word? Davis demonstrates how mathematics surrounds, imbues, and maintains our everyday lives: the digitization and automation of processes like pumping gas, withdrawing cash, and buying groceries are all fueled by mathematics. He takes the reader through a point-by-point explanation of many frequently asked questions about mathematics, gently introducing this Handmaiden of Science and telling you everything you've ever wanted to know about her.

"In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began."-Albert Einstein The year was 1915, and the young mathematician Emmy Noether had just settled into Gottingen University when Albert Einstein visited to lecture on his nearly finished general theory of relativity. Two leading mathematicians of the day, David Hilbert and Felix Klein, dug into the new theory with gusto, but had difficulty reconciling it with what was known about the conservation of energy. Knowing of her expertise in invariance theory, they requested Noether's help. To solve the problem, she developed a novel theorem, applicable across all of physics, which relates conservation laws to continuous symmetries-one of the most important pieces of mathematical reasoning ever developed. Noether's "first" and "second" theorem was published in 1918. The first theorem relates symmetries under global spacetime transformations to the conservation of energy and momentum, and symmetry under global gauge transformations to charge conservation. In continuum mechanics and field theories, these conservation laws are expressed as equations of continuity. The second theorem, an extension of the first, allows transformations with local gauge invariance, and the equations of continuity acquire the covariant derivative characteristic of coupled matter-field systems. General relativity, it turns out, exhibits local gauge invariance. Noether's theorem also laid the foundation for later generations to apply local gauge invariance to theories of elementary particle interactions. In Dwight E. Neuenschwander's new edition of Emmy Noether's Wonderful Theorem, readers will encounter an updated explanation of Noether's "first" theorem. The discussion of local gauge invariance has been expanded into a detailed presentation of the motivation, proof, and applications of the "second" theorem, including Noether's resolution of concerns about general relativity. Other refinements in the new edition include an enlarged biography of Emmy Noether's life and work, parallels drawn between the present approach and Noether's original 1918 paper, and a summary of the logic behind Noether's theorem.

This book is a consequence of the international meeting organized in Marseilles in November 2018 devoted to the aftermath of the Great War for mathematical communities. It features selected original research presented at the meeting offering a new perspective on a period, the 1920s, not extensively considered by historiography. After 1918, new countries were created, and borders of several others were modified. Territories were annexed while some countries lost entire regions. These territorial changes bear witness to the massive and varied upheavals with which European societies were confronted in the aftermath of the Great War. The reconfiguration of political Europe was accompanied by new alliances and a redistribution of trade - commercial, intellectual, artistic, military, and so on - which largely shaped international life during the interwar period. These changes also had an enormous impact on scientific life, not only in practice, but also in its organization and communication strategies. The mathematical sciences, which from the late 19th century to the 1920s experienced a deep disciplinary evolution, were thus facing a double movement, internal and external, which led to a sustainable restructuring of research and teaching. Concomitantly, various areas such as topology, functional analysis, abstract algebra, logic or probability, among others, experienced exceptional development. This was accompanied by an explosion of new international or national associations of mathematicians with for instance the founding, in 1918, of the International Mathematical Union and the controversial creation of the International Research Council. Therefore, the central idea for the articulation of the various chapters of the book is to present case studies illustrating how in the aftermath of the war, many mathematicians had to organize their personal trajectories taking into account the evolution of the political, social and scientific environment which had taken place at the end of the conflict.

Symbols, Impossible Numbers, and Geometric Entanglements is the first history of the development and reception of algebra in early modern England and Scotland. Not primarily a technical history, this book analyses the struggles of a dozen British thinkers to come to terms with early modern algebra, its symbolic style, and negative and imaginary numbers. Professor Pycior uncovers these thinkers as a 'test-group' for the symbolic reasoning that would radically change not only mathematics but also logic, philosophy and language studies. The book furthermore shows how pedagogical and religious concerns shaped the British debate over the relative merits of algebra and geometry. Positioning algebra firmly in the Scientific Revolution and pursue Newton the algebraist, it highlights Newton's role in completing the evolution of algebra from an esoteric subject into a major focus of British mathematics. Other thinkers covered include Oughtred, Harriot, Wallis, Hobbes, Barrow, Berkeley and MacLaurin.

The Development of Mathematics in Medieval Europe complements the previous collection of articles by Menso Folkerts, Essays on Early Medieval Mathematics, and deals with the development of mathematics in Europe from the 12th century to about 1500. In the 12th century European learning was greatly transformed by translations from Arabic into Latin. Such translations in the field of mathematics and their influence are here described and analysed, notably al-Khwarizmi's "Arithmetic" -- through which Europe became acquainted with the Hindu-Arabic numerals -- and Euclid's "Elements". Five articles are dedicated to Johannes Regiomontanus, perhaps the most original mathematician of the 15th century, and to his discoveries in trigonometry, algebra and other fields. The knowledge and application of Euclid's "Elements" in 13th- and 15th-century Italy are discussed in three studies, while the last article treats the development of algebra in South Germany around 1500, where much of the modern symbolism used in algebra was developed.

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

The studies brought together in this second collection of articles by Paul Kunitzsch continue the lines of research evident in his previous volume (The Arabs and the Stars). The Arabic materials discussed stem mostly from the early period of the development of Arabic-Islamic astronomy up to about 1000AD, while the Latin materials belong to the first stage of Western contact with Arabic science at the end of the 10th century, and to the peak of Arabic-Latin translation activity in 12th century Spain. The first set of articles focuses upon Ptolemy in the Arabic-Latin tradition, followed by further ones on Arabic astronomy and its reception in the West; the final group looks at details of the transmission of Euclid's Elements.

Jamshid al-Kashi's Miftah al-Hisab (Key to Arithmetic) was largely unknown to researchers until the mid-20th century, and has not been translated to English until now. This is the second book in a multi-volume set that finally brings al-Kashi's groundbreaking textbook to English audiences in its entirety. As soon as it was studied by modern researchers, it changed some false assumptions about the history of certain topics in mathematics. Written as a textbook for students of mathematics, astronomy, accounting, engineering, and architecture, Miftah covers a wide range of topics in arithmetic, geometry, and algebra. By sharing al-Kashi's most comprehensive work with a wider audience, this book will help establish a more complete history of mathematics, and extend al-Kashi's influence into the 21st century and beyond. The book opens by briefly recounting al-Kashi's biography, so as to situate readers in the work's rich historical context. His impressive status in the kingdom of Ulugh Beg is detailed, as well as his contributions to both mathematics and astronomy. As a master calculator and astronomer, al-Kashi's calculations of 2 and sin(10) were by far the most accurate for almost two centuries. His law of cosines is still studied in schools today. This translation contributes to the understanding and appreciation of al-Kashi's esteemed place in the scientific world. A side-by-side presentation of the source manuscript-one of the oldest known copies-and the English translation is provided on each page. Detailed footnotes are also provided throughout, which will offer readers an even deeper look at the text's mathematical and historical basis. Researchers and students of the history of mathematics will find this volume indispensable in filling in a frequently overlooked time period and region. This volume will also provide anybody interested in the history of Islamic culture with an insightful look at one of the mathematical world's most neglected figures.

This handbook features essays written by both literary scholars and mathematicians that examine multiple facets of the connections between literature and mathematics. These connections range from mathematics and poetic meter to mathematics and modernism to mathematics as literature. Some chapters focus on a single author, such as mathematics and Ezra Pound, Gertrude Stein, or Charles Dickens, while others consider a mathematical topic common to two or more authors, such as squaring the circle, chaos theory, Newton's calculus, or stochastic processes. With appeal for scholars and students in literature, mathematics, cultural history, and history of mathematics, this important volume aims to introduce the range, fertility, and complexity of the connections between mathematics, literature, and literary theory. Chapter 1 is available open access under a Creative Commons Attribution 4.0 International License via [link.springer.com|http://link.springer.com/].

The great mathematician Archimedes, a Sicilian Greek whose machines defended Syracuse against the Romans during the Second Punic War, was killed by a Roman after the city fell, yet it is largely Roman sources, and Greek texts aimed at Roman audiences, that preserve the stories about him. Archimedes' story, Mary Jaeger argues, thus becomes a locus where writers explore the intersection of Greek and Roman culture, and as such it plays an important role in Roman self-definition. Jaeger uses the biography of Archimedes as a hermeneutic tool, providing insight into the construction of the traditional historical narrative about the Roman conquest of the Greek world and the Greek cultural invasion of Rome.

By breaking down the narrative of Archimedes' life and examining how the various anecdotes that comprise it are embedded in their contexts, the book offers fresh readings of passages from both well-known and less-studied authors, including Polybius, Cicero, Livy, Vitruvius, Plutarch, Silius Italicus, Valerius Maximus, Johannes Tzetzes, and Petrarch.

This book focuses on the ancient Near East, early imperial China, South-East Asia, and medieval Europe, shedding light on mathematical knowledge and practices documented by sources relating to the administrative and economic activities of officials, merchants and other actors. It compares these to mathematical texts produced in related school contexts or reflecting the pursuit of mathematics for its own sake to reveal the diversity of mathematical practices in each of these geographical areas of the ancient world. Based on case studies from various periods and political, economic and social contexts, it explores how, in each part of the world discussed, it is possible to identify and describe the different cultures of quantification and computation as well as their points of contact. The thirteen chapters draw on a wide variety of texts from ancient Near East, China, South-East Asia and medieval Europe, which are analyzed by researchers from various fields, including mathematics, history, philology, archaeology and economics. The book will appeal to historians of science, economists and institutional historians of the ancient and medieval world, and also to Assyriologists, Indologists, Sinologists and experts on medieval Europe.

During Song (960 to 1279) and Yuan (1279 to 1368) dynasties, China experienced a peak in high-level algebraic investigation through the works of famous mathematicians such as Qin Jiushao, Zhu Shijie, Yang Hui and Li Ye. Among these is Li Ye's short treatise on a curious ancient geometrical procedure: The Development of Pieces of Areas According to the Collection Augmenting the Ancient Knowledge (Yigu yanduan). The aim of this monography is to contradict traditional scholarship which has long discredited the importance of Li Ye's treatise, considering it a mere popular handbook. The author aims to show that Li Ye's work actually epitomizes a completely new aspect of ancient Chinese mathematics: a crossroad between algebra, geometry, and combinatorics containing elements reminiscent of the Book of Changes (Yi Jing). As well as Li Ye used field measurement as pretext for investigations on quadratic equations and Changes, the present study uses Li Ye's small treatise as pretext for philosophical investigations on link between mathematics and their history. The real topic of the study is the exploration of another expression of proof and generality in Chinese mathematics. This book not only completes the edition of Li Ye's works and presents new features of Chinese mathematics, but also fills a gap in the translation of Chinese mathematics treatises.It is the first book entirely dedicated to the diagrammatic practice of algebra in the history of Chinese mathematics. This practice is more important than expected. While being a monograph, the book is short and detailed enough to be used by students in class. It can also be used as an entry door to the research field of history of Chinese mathematics.

This volume offers an integrated understanding of how the theory of general relativity gained momentum after Einstein had formulated it in 1915. Chapters focus on the early reception of the theory in physics and philosophy and on the systematic questions that emerged shortly after Einstein's momentous discovery. They are written by physicists, historians of science, and philosophers, and were originally presented at the conference titled Thinking About Space and Time: 100 Years of Applying and Interpreting General Relativity, held at the University of Bern from September 12-14, 2017. By establishing the historical context first, and then moving into more philosophical chapters, this volume will provide readers with a more complete understanding of early applications of general relativity (e.g., to cosmology) and of related philosophical issues. Because the chapters are often cross-disciplinary, they cover a wide variety of topics related to the general theory of relativity. These include: Heuristics used in the discovery of general relativity Mach's Principle The structure of Einstein's theory Cosmology and the Einstein world Stability of cosmological models The metaphysical nature of spacetime The relationship between spacetime and dynamics The Geodesic Principle Symmetries Thinking About Space and Time will be a valuable resource for historians of science and philosophers who seek a deeper knowledge of the (early and later) uses of general relativity, as well as for physicists and mathematicians interested in exploring the wider historical and philosophical context of Einstein's theory.

Medieval Islamic World: An Intellectual History of Science and Politics surveys major scientific and philosophical discoveries in the medieval period within the broader Islamicate world, providing an alternative historical framework to that of the primarily Eurocentric history of science and philosophy of science and technology fields. Medieval Islamic World serves to address the history of rationalist inquiry within scholarly institutions in medieval Islamic societies, surveying developments in the fields of medicine and political theory, and the scientific disciplines of astronomy, chemistry, physics, and mechanics, as led by medieval Muslim scholarship.

About Felix Klein, the famous Greek mathematician Constantin Caratheodory once said: "It is only by illuminating him from all angles that one can come to understand his significance." The author of this biography has done just this. A detailed study of original sources has made it possible to uncover new connections; to create a more precise representation of this important mathematician, scientific organizer, and educational reformer; and to identify misconceptions. Because of his edition of Julius Plucker's work on line geometry and due to his own contributions to non-Euclidean geometry, Klein was already well known abroad before he received his first full professorship at the age of 23. By exchanging ideas with his most important cooperation partner, the Norwegian Sophus Lie, Klein formulated his Erlangen Program. Various other visionary programs followed, in which Klein involved mathematicians from Germany and abroad. Klein was the most active promoter of Riemann's geometric-physical approach to function theory, but he also integrated the analytical approaches of the Weierstrass school into his arsenal of methods. Klein was a citizen of the world who repeatedly travelled to France, Great Britain, Italy, the United States, and elsewhere. Despite what has often been claimed, it must be emphasized that Klein expressly opposed national chauvinism. He promoted mathematically gifted individuals regardless of their nationality, religion, or gender. Many of his works have been translated into English, French, Italian, Russian, and other languages; more than 300 supporters from around the world made it possible for his portrait to be painted by the prominent impressionist Max Liebermann. Inspired by international developments, Klein paved the way for women to work in the field of mathematics. He was instrumental in reforming mathematical education, and he endorsed an understanding of mathematics that affirmed its cultural importance as well as its fundamental significance to scientific and technological progress.

By the end of the 1960s, a new discipline named computer science had come into being. A new scientific paradigm-the 'computational paradigm'-was in place, suggesting that computer science had reached a certain level of maturity. Yet as a science it was still precociously young. New forces, some technological, some socio-economic, some cognitive impinged upon it, the outcome of which was that new kinds of computational problems arose over the next two decades. Indeed, by the beginning of the 1990's the structure of the computational paradigm looked markedly different in many important respects from how it was at the end of the 1960s. Author Subrata Dasgupta named the two decades from 1970 to 1990 as the second age of computer science to distinguish it from the preceding genesis of the science and the age of the Internet/World Wide Web that followed. This book describes the evolution of computer science in this second age in the form of seven overlapping, intermingling, parallel histories that unfold concurrently in the course of the two decades. Certain themes characteristic of this second age thread through this narrative: the desire for a genuine science of computing; the realization that computing is as much a human experience as it is a technological one; the search for a unified theory of intelligence spanning machines and mind; the desire to liberate the computational mind from the shackles of sequentiality; and, most ambitiously, a quest to subvert the very core of the computational paradigm itself. We see how the computer scientists of the second age address these desires and challenges, in what manner they succeed or fail and how, along the way, the shape of computational paradigm was altered. And to complete this history, the author asks and seeks to answer the question of how computer science shows evidence of progress over the course of its second age.

How can we invent new certain knowledge in a methodical manner? This question stands at the heart of Salomon Maimon's theory of invention. Chikurel argues that Maimon's contribution to the ars inveniendi tradition lies in the methods of invention which he prescribes for mathematics. Influenced by Proclus' commentary on Elements, these methods are applied on examples taken from Euclid's Elements and Data. Centering around methodical invention and scientific genius, Maimon's philosophy is unique in an era glorifying the artistic genius, known as Geniezeit. Invention, primarily defined as constructing syllogisms, has implications on the notion of being given in intuition as well as in symbolic cognition. Chikurel introduces Maimon's notion of analysis in the broader sense, grounded not only on the principle of contradiction but on intuition as well. In philosophy, ampliative analysis is based on Maimon's logical term of analysis of the object, a term that has yet to be discussed in Maimonian scholarship. Following its introduction, a new version of the question quid juris? arises. In mathematics, Chikurel demonstrates how this conception of analysis originates from practices of Greek geometrical analysis.

This monograph examines the private annotations that Ludwig Wittgenstein made to his copy of G.H. Hardy's classic textbook, A Course of Pure Mathematics. Complete with actual images of the annotations, it gives readers a more complete picture of Wittgenstein's remarks on irrational numbers, which have only been published in an excerpted form and, as a result, have often been unjustly criticized. The authors first establish the context behind the annotations and discuss the historical role of Hardy's textbook. They then go on to outline Wittgenstein's non-extensionalist point of view on real numbers, assessing his manuscripts and published remarks and discussing attitudes in play in the philosophy of mathematics since Dedekind. Next, coverage focuses on the annotations themselves. The discussion encompasses irrational numbers, the law of excluded middle in mathematics and the notion of an "improper picture," the continuum of real numbers, and Wittgenstein's attitude toward functions and limits.

Drawing on published works, correspondence and manuscripts, this book offers the most comprehensive reconstruction of Boscovich's theory within its historical context. It explains the genesis and theoretical as well as epistemological underpinnings in light of the Jesuit tradition to which Boscovich belonged, and contrasts his ideas with those of Newton, Leibniz, and their legacy. Finally, it debates crucial issues in early-modern physical science such as the concept of force, the particle-like structure of matter, the idea of material points and the notion of continuity, and shares novel insights on Boscovich's alleged influence on later developments in physics. With its attempt to reduce all natural forces to one single law, Boscovich's Theory of Natural Philosophy, published in 1758, left a lasting impression on scientists and philosophers of every age regarding the fundamental unity of physical phenomena. The theory argues that every pair of material points is subject to one mutual force - and always the same force - which is their propensity to be mutually attracted or repelled, depending on their distance from one another. Furthermore, the action of this unique force is visualized through a famous diagram that fascinated generations of scientists. But his understanding of key terms of the theory - such as the notion of force involved and the very idea of a material point - is only ostensibly similar to our current conceptual framework. Indeed, it needs to be clarified within the plurality of contexts in which it has emerged rather than being considered in view of later developments. The book is recommended for scholars and students interested in the ideas of the early modern period, especially historians and philosophers of science, mathematicians and physicists with an interest in the history of the discipline, and experts on Jesuit science and philosophy.

This book presents, in his own words, the life of Hugo Steinhaus (1887-1972), noted Polish mathematician of Jewish background, educator, and mathematical popularizer. A student of Hilbert, a pioneer of the foundations of probability and game theory, and a contributor to the development of functional analysis, he was one of those instrumental to the extraordinary flowering of Polish mathematics before and after World War I. In particular, it was he who "discovered" the great Stefan Banach. Exhibiting his great integrity and wit, Steinhaus's personal story of the turbulent times he survived - including two world wars and life postwar under the Soviet heel - cannot but be of consuming interest. His recounting of the fearful years spent evading Nazi terror is especially moving. The steadfast honesty and natural dignity he maintained while pursuing a life of demanding scientific and intellectual enquiry in the face of encroaching calamity and chaos show him to be truly a mathematician for all seasons. The present work will be of great interest not only to mathematicians wanting to learn some of the details of the mathematical blossoming that occurred in Poland in the first half of the 20th century, but also to anyone wishing to read a first-hand account of the history of those unquiet times in Europe - and indeed world-wide - by someone of uncommon intelligence and forthrightness situated near an eye of the storm.

This book documents the process of transformation from natural philosophy, which was considered the most important of the sciences until the early modern era, into modern disciplines such as mathematics, physics, natural history, chemistry, medicine and engineering. It focuses on the 18th century, which has often been considered uninteresting for the history of science, representing the transition from the age of genius and the birth of modern science (the 17th century) to the age of prodigious development in the 19th century. Yet the 18th century, the century of Enlightenment, as will be demonstrated here, was in fact characterized by substantial ferment and novelty. To make the text more accessible, little emphasis has been placed on the precise genesis of the various concepts and methods developed in scientific enterprises, except when doing so was necessary to make them clear. For the sake of simplicity, in several situations reference is made to the authors who are famous today, such as Newton, the Bernoullis, Euler, d'Alembert, Lagrange, Lambert, Volta et al. - not necessarily because they were the most creative and original minds, but mainly because their writings represent a synthesis of contemporary and past studies. The above names should, therefore, be considered more labels of a period than references to real historical characters.

This book presents the classical theory of curves in the plane and three-dimensional space, and the classical theory of surfaces in three-dimensional space. It pays particular attention to the historical development of the theory and the preliminary approaches that support contemporary geometrical notions. It includes a chapter that lists a very wide scope of plane curves and their properties. The book approaches the threshold of algebraic topology, providing an integrated presentation fully accessible to undergraduate-level students.

At the end of the 17th century, Newton and Leibniz developed differential calculus, thus making available the very wide range of differentiable functions, not just those constructed from polynomials. During the 18th century, Euler applied these ideas to establish what is still today the classical theory of most general curves and surfaces, largely used in engineering. Enter this fascinating world through amazing theorems and a wide supply of surprising examples. Reach the doors of algebraic topology by discovering just how an integer (= the Euler-Poincare characteristics) associated with a surface gives you a lot of interesting information on the shape of the surface. And penetrate the intriguing world of Riemannian geometry, the geometry that underlies the theory of relativity.

The book is of interest to all those who teach classical differential geometry up to quite an advanced level. The chapter on Riemannian geometry is of great interest to those who have to intuitively introduce students to the highly technical nature of this branch of mathematics, in particular when preparing students for courses on relativity."