A member of the Academie francaise, Henri Poincare (1854 1912) was one of the greatest mathematicians and theoretical physicists of the late nineteenth and early twentieth centuries. His discovery of chaotic motion laid the foundations of modern chaos theory, and he was acknowledged by Einstein as a key contributor in the field of special relativity. He earned his enduring reputation as a philosopher of mathematics and science with this elegantly written work, which was first published in French as three separate essays: Science and Hypothesis (1902), The Value of Science (1905), and Science and Method (1908). Poincare asserts that much scientific work is a matter of convention, and that intuition and prediction play key roles. George Halsted's authorised 1913 English translation retains Poincare's lucid prose style, presenting complex ideas for both professional scientists and those readers interested in the history of mathematics and the philosophy of science."

This is a comprehensive book on the life and works of Leon Henkin (1921-2006), an extraordinary scientist and excellent teacher whose writings became influential right from the beginning of his career with his doctoral thesis on "The completeness of formal systems" under the direction of Alonzo Church. Upon the invitation of Alfred Tarski, Henkin joined the Group in Logic and the Methodology of Science in the Department of Mathematics at the University of California Berkeley in 1953. He stayed with the group until his retirement in 1991. This edited volume includes both foundational material and a logic perspective. Algebraic logic, model theory, type theory, completeness theorems, philosophical and foundational studies are among the topics covered, as well as mathematical education. The work discusses Henkin's intellectual development, his relation to his predecessors and contemporaries and his impact on the recent development of mathematical logic. It offers a valuable reference work for researchers and students in the fields of philosophy, mathematics and computer science.

A follow-up to the volume "Discovering the Principles of Mechanics 1600-1800. Essays by David Speiser" (Birkhauser 2008), this volume contains the essays of David Speiser on relationships between science, history of science, history of art and philosophy.

This volume commemorates the life, work and foundational views of Kurt Goedel (1906-78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Goedel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Goedel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Throughout his early life, Isaac Todhunter (1820-84) excelled as a student of mathematics, gaining a scholarship at the University of London and numerous awards during his time at St John's College, Cambridge. Taking up fellowship of the college in 1849, he became widely known for both his educational texts and his historical accounts of various branches of mathematics. The present work, first published in 1865, describes the rise of probability theory as a recognised subject, beginning with a discussion of the famous 'problem of points', as considered by the likes of the Chevalier de Mere, Blaise Pascal and Pierre de Fermat during the latter half of the seventeenth century. Subsequently, the application of advanced methods that had been developed in classical areas of mathematics led to rapid progress in probability theory. Todhunter traces this growth, closing with a thorough account of Pierre-Simon Laplace's far-reaching work in the area.

Successful long-distance navigation depends on knowing latitude and longitude, and the determination of longitude depends on knowing the exact time at some fixed point on the earth's surface. Since Newton it had been hoped that a method based on accurate prediction of the moon's orbit would give such a time. Building on the work of Euler, Thomas Mayer and others, the astronomer and mathematician Nevil Maskelyne (1732-1811) was able to devise such a method and yearly publication of the Nautical Almanac and Astronomical Ephemeris placed it in the hands of every ship's captain. First published in 1767 and reissued here in the revised third edition of 1802, the present work provided the necessary tables and instructions. The development of rugged and accurate chronometers eventually displaced Maskelyne's method, but navigators continued to make use of it for many decades. This edition of the tables notably formed part of the library of the Beagle on Darwin's famous voyage.

This book offers an accessible and in-depth look at some of the most important episodes of two thousand years of mathematical history. Beginning with trigonometry and moving on through logarithms, complex numbers, infinite series, and calculus, this book profiles some of the lesser known but crucial contributors to modern day mathematics. It is unique in its use of primary sources as well as its accessibility; a knowledge of first-year calculus is the only prerequisite. But undergraduate and graduate students alike will appreciate this glimpse into the fascinating process of mathematical creation. The history of math is an intercontinental journey, and this book showcases brilliant mathematicians from Greece, Egypt, and India, as well as Europe and the Islamic world. Several of the primary sources have never before been translated into English. Their interpretation is thorough and readable, and offers an excellent background for teachers of high school mathematics as well as anyone interested in the history of math.

From the end of antiquity to the middle of the nineteenth century it was generally believed that Aristotle had said all that there was to say concerning the rules of logic and inference. One of the ablest British mathematicians of his age, Augustus De Morgan (1806-71) played an important role in overturning that assumption with the publication of this book in 1847. He attempts to do several things with what we now see as varying degrees of success. The first is to treat logic as a branch of mathematics, more specifically as algebra. Here his contributions include his laws of complementation and the notion of a universe set. De Morgan also tries to tie together formal and probabilistic inference. Although he is never less than acute, the major advances in probability and statistics at the beginning of the twentieth century make this part of the book rather less prophetic.

A miller's son, George Green (1793 1841) received little formal schooling yet managed to acquire significant knowledge of modern mathematics, especially French work. In 1828 he published his Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, the work for which he is now celebrated. Admitted to Cambridge in 1833 as a mature student, Green went on to become a fellow of Gonville and Caius College. His early death, however, cut short a promising career as a mathematical physicist. While English contemporaries saw what he might have achieved, they did not understand what he had actually achieved. Only when William Thomson (later Lord Kelvin) rediscovered Green's first publication and shared it with the French mathematical elite was his greatness truly appreciated. Edited by the Cambridge mathematician Norman Macleod Ferrers (1829 1903) and published in 1871, this collection comprises Green's influential essay and nine further papers."

The scientific personalities of Luigi Cremona, Eugenio Beltrami, Salvatore Pincherle, Federigo Enriques, Beppo Levi, Giuseppe Vitali, Beniamino Segre and of several other mathematicians who worked in Bologna in the century 1861-1960 are examined by different authors, in some cases providing different view points. Most contributions in the volume are historical; they are reproductions of original documents or studies on an original work and its impact on later research. The achievements of other mathematicians are investigated for their present-day importance.

In his autobiography, Charles Darwin wrote of his time at Cambridge: 'I attempted mathematics ... but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.' First published in 1795 and reissued here in its 1815 sixth edition, The Elements of Algebra by James Wood (1760-1839) was one of the standard Cambridge texts for decades, so its presence in Darwin's library aboard the Beagle is readily understandable. Then, as now, Cambridge had a high opinion of itself as a mathematical university. The contents of Wood's book give an interesting glimpse of the standards expected of the less able students.

Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and is best-known for the first formulation of the axiom of choice. However, his papers also include pioneering work in applied mathematics and mathematical physics. This edition of his collected papers consists of two volumes. The present Volume II covers Ernst Zermelo's work on the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field who comments on the historical background, motivation, accomplishments, and influence.

This book is a collection of essays on the reception of Leibniz's thinking in the sciences and in the philosophy of science in the 19th and 20th centuries. Authors studied include C.F. Gauss, Georg Cantor, Kurd Lasswitz, Bertrand Russell, Ernst Cassirer, Louis Couturat, Hans Reichenbach, Hermann Weyl, Kurt Goedel and Gregory Chaitin. In addition, we consider concepts and problems central to Leibniz's thought and that of the later authors: the continuum, space, identity, number, the infinite and the infinitely small, the projects of a universal language, a calculus of logic, a mathesis universalis etc. The book brings together two fields of research in the history of philosophy and of science (research on Leibniz, and the research concerned with some major developments in the 19th and 20th centuries); it describes how Leibniz's thought appears in the works of these authors, in order to better understand Leibniz's influence on contemporary science and philosophy; but it also assesses that reception critically, confronting it in particular with the current state of Leibniz research and with the various editions of his work.

The Swiss mathematician Jakob Steiner (1796-1863) came from a poor background with an incomplete education, yet such was his mathematical talent that eventually the Prussian university system adapted itself to him rather than he to it. A geometer in an age dominated by analysts, he pursued his own interests in his own way. The elegant results which bear his name - including Steiner circles, systems and symmetrisation - are known to most mathematicians today. Considered by many to be the greatest geometer since Apollonius of Perga, Steiner did important work on systemising geometry, laying the foundation for much later work on projective geometry. Edited by the eminent mathematician Karl Weierstrass (1815-97), this two-volume edition of Steiner's collected works offers scholars access to his influential writings in the original German. Volume 1 was published in 1881.

By the end of the eighteenth century, British mathematics had been stuck in a rut for a hundred years. Calculus was still taught in the style of Newton, with no recognition of the great advances made in continental Europe. The examination system at Cambridge even mandated the use of Newtonian notation. As discontented undergraduates, Charles Babbage (1791 1871) and John Herschel (1792 1871) formed the Analytical Society in 1811. The group, including William Whewell and George Peacock, sought to promote the new continental mathematics. Babbage's preface to the present work, first published in 1813, may be considered the movement's manifesto. He provided the first paper here, and Herschel the two others. Although the group was relatively short-lived, its ideas took root as its erstwhile members rose to prominence. As the society's sole publication, this remains a significant text in the history of British mathematics.

A distinguished mathematician and notable university teacher, Isaac Todhunter (1820 84) became known for the successful textbooks he produced as well as for a work ethic that was extraordinary, even by Victorian standards. A scholar who read all the major European languages, Todhunter was an open-minded man who admired George Boole and helped introduce the moral science examination at Cambridge. His many gifts enabled him to produce the histories of mathematical subjects which form his lasting memorial. First published between 1886 and 1893, the present work was the last of these. Edited and completed after Todhunter's death by Karl Pearson (1857 1936), another extraordinary man who pioneered modern statistics, these volumes trace the mathematical understanding of elasticity from the seventeenth to the late nineteenth century. Volume 2 (1893) was split into two parts. Part 2 covers the work of Neumann, Kirchhoff, Clebsch, Boussinesq, and Lord Kelvin."

A distinguished mathematician and notable university teacher, Isaac Todhunter (1820 84) became known for the successful textbooks he produced as well as for a work ethic that was extraordinary, even by Victorian standards. A scholar who read all the major European languages, Todhunter was an open-minded man who admired George Boole and helped introduce the moral science examination at Cambridge. His many gifts enabled him to produce the histories of mathematical subjects which form his lasting memorial. First published between 1886 and 1893, the present work was the last of these. Edited and completed after Todhunter's death by Karl Pearson (1857 1936), another extraordinary man who pioneered modern statistics, these volumes trace the mathematical understanding of elasticity from the seventeenth to the late nineteenth century. Volume 2 (1893) was split into two parts. Part 1 includes the work of Saint-Venant from 1850 to 1886."

A distinguished mathematician and notable university teacher, Isaac Todhunter (1820 84) became known for the successful textbooks he produced as well as for a work ethic that was extraordinary, even by Victorian standards. A scholar who read all the major European languages, Todhunter was an open-minded man who admired George Boole and helped introduce the moral science examination at Cambridge. His many gifts enabled him to produce the histories of mathematical subjects which form his lasting memorial. First published between 1886 and 1893, the present work was the last of these. Edited and completed after Todhunter's death by Karl Pearson (1857 1936), another extraordinary man who pioneered modern statistics, these volumes trace the mathematical understanding of elasticity from the seventeenth to the late nineteenth century. Volume 1 (1886) begins with Galileo Galilei and extends to the researches of Saint-Venant up to 1850."

The Belgian polymath Lambert Adolphe Jacques Quetelet (1796-1874) was regarded by John Maynard Keynes as a 'parent of modern statistical method'. Applying his training in mathematics to the physical and psychological dimensions of individuals, his Treatise on Man (also reissued in this series) identified the 'average man' in statistical terms. Reissued here is the 1839 English translation of his 1828 work, which appeared at a time when the application of probability was moving away from gaming tables towards more useful areas of life. Quetelet believed that probability had more influence on human affairs than had been accepted, and this work marked his move from a focus on mathematics and the natural sciences to the study of statistics and, eventually, the investigation of social phenomena. Written as a summary of lectures given in Brussels, the work was translated from French by the engineer Richard Beamish (1798-1873).

Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). The book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.

'If one would understand the Greek genius fully, it would be a good plan to begin with their geometry.' As early as the sixth century BCE, Thales of Miletus used geometrical principles to calculate distance and height. Within a few hundred years, Euclid had produced his seminal Elements, which was still used as a textbook when this two-volume work was first published in 1921. A distinguished civil servant as well as an expert on ancient Greek mathematics, Sir Thomas Little Heath (1861 1940) includes here sufficient detail for a modern mathematician to grasp ancient methodology, alongside explanatory sections aimed at classicists. This remains a rigorous and essential exposition of a vast topic. Volume 2 focuses on post-Euclidian mathematics, beginning with the work of Aristarchus of Samos and extending to that of Diophantus of Alexandria. Heath had previously published separate studies on these two thinkers (also reissued in this series)."

Active in Alexandria in the third century BCE, Apollonius of Perga ranks as one of the greatest Greek geometers. Building on foundations laid by Euclid, he is famous for defining the parabola, hyperbola and ellipse in his major treatise on conic sections. The dense nature of its text, however, made it inaccessible to most readers. When it was originally published in 1896 by the civil servant and classical scholar Thomas Little Heath (1861 1940), the present work was the first English translation and, more importantly, the first serious effort to standardise the terminology and notation. Along with clear diagrams, Heath includes a thorough introduction to the work and the history of the subject. Seeing the treatise as more than an esoteric artefact, Heath presents it as a valuable tool for modern mathematicians. His works on Diophantos of Alexandria (1885) and Aristarchus of Samos (1913) are also reissued in this series.

'If one would understand the Greek genius fully, it would be a good plan to begin with their geometry.' As early as the sixth century BCE, Thales of Miletus used geometrical principles to calculate distance and height. Within a few hundred years, Euclid had produced his seminal Elements, which was still used as a textbook when this two-volume work was first published in 1921. A distinguished civil servant as well as an expert on ancient Greek mathematics, Sir Thomas Little Heath (1861 1940) includes here sufficient detail for a modern mathematician to grasp ancient methodology, alongside explanatory sections aimed at classicists. This remains a rigorous and essential exposition of a vast topic. Volume 1 includes an introduction that touches on the conditions which made possible the rapid development of philosophy and science in ancient Greece. The coverage begins with Thales and ends with Euclid."

These three volumes constitute the first complete English translation of Felix Klein's seminal series "Elementarmathematik vom hoheren Standpunkte aus". "Complete" has a twofold meaning here: First, there now exists a translation of volume III into English, while until today the only translation had been into Chinese. Second, the English versions of volume I and II had omitted several, even extended parts of the original, while we now present a complete revised translation into modern English. The volumes, first published between 1902 and 1908, are lecture notes of courses that Klein offered to future mathematics teachers, realizing a new form of teacher training that remained valid and effective until today: Klein leads the students to gain a more comprehensive and methodological point of view on school mathematics. The volumes enable us to understand Klein's far-reaching conception of elementarisation, of the "elementary from a higher standpoint", in its implementation for school mathematics. This volume I is devoted to what Klein calls the three big "A's": arithmetic, algebra and analysis. They are presented and discussed always together with a dimension of geometric interpretation and visualisation - given his epistemological viewpoint of mathematics being based in space intuition. A particularly revealing example for elementarisation is his chapter on the transcendence of e and p, where he succeeds in giving concise yet well accessible proofs for the transcendence of these two numbers. It is in this volume that Klein makes his famous statement about the double discontinuity between mathematics teaching at schools and at universities - it was his major aim to overcome this discontinuity.

When George Shoobridge Carr (1837-1914) wrote his Synopsis of Elementary Results he intended it as an aid to students preparing for degree-level examinations such as the Cambridge Mathematical Tripos, for which he provided private tuition. He would have been startled to see the two volumes, first published in 1880 and 1886 respectively, reissued more than a century later. Notably, in 1903 the work fell into the hands of the Indian prodigy Srinivasa Ramanujan (1887-1920) and greatly influenced his mathematical education. It is the interaction between a methodical teaching aid and the soaring spirit of a self-taught genius which gives this reissue its interest. Volume 2 contains sections on differential calculus, integral calculus, calculus of variations, differential equations, calculus of finite differences, plane coordinate geometry and solid coordinate geometry. Also included is a historically valuable index insofar as it provides references to 890 volumes of 32 periodicals dating back to 1800.