Great mathematicians write for the future and Georg Friedrich Bernhard Riemann (1826-66) was one of the greatest mathematicians of all time. Edited by Heinrich Martin Weber, with assistance from Richard Dedekind, this edition of his collected works in German first appeared in 1876. Riemann's interests ranged from pure mathematics to mathematical physics. He wrote a short paper on number theory which provided the key to the prime number theorem, and his zeta hypothesis has given mathematicians the most famous of today's unsolved problems. Moreover, his famous 1854 lecture 'On the hypotheses which underlie geometry' set in motion studies which culminated in Einstein's general theory of relativity. Even Riemann's over-optimistic use of the Dirichlet principle to prove the conformal mapping theorem turned out to be immensely fruitful. The alert reader will further profit from finding here the seeds of modern distribution theory, algebraic topology and measure theory.

The amount of mathematics invented for number-theoretic reasons is impressive. It includes much of complex analysis, the re-foundation of algebraic geometry on commutative algebra, group cohomology, homological algebra, and the theory of motives. Zeta and L-functions sit at the meeting point of all these theories and have played a profound role in shaping the evolution of number theory. This book presents a big picture of zeta and L-functions and the complex theories surrounding them, combining standard material with results and perspectives that are not made explicit elsewhere in the literature. Particular attention is paid to the development of the ideas surrounding zeta and L-functions, using quotes from original sources and comments throughout the book, pointing the reader towards the relevant history. Based on an advanced course given at Jussieu in 2013, it is an ideal introduction for graduate students and researchers to this fascinating story.

The German mathematician Karl Weierstrass (1815-97) is generally considered to be the father of modern analysis. His clear eye for what was important is demonstrated by the publication, late in life, of his polynomial approximation theorem; suitably generalised as the Stone-Weierstrass theorem, it became a central tool for twentieth-century analysis. Furthermore, the Weierstrass nowhere-differentiable function is the seed from which springs the entire modern theory of mathematical finance. The best students in Europe came to Berlin to attend his lectures, and his rigorous style still dominates the first analysis course at any university. His seven-volume collected works in the original German contain not only published treatises but also records of many of his famous lecture courses. Volume 1 was published in 1894.

The German mathematician Karl Weierstrass (1815-97) is generally considered to be the father of modern analysis. His clear eye for what was important is demonstrated by the publication, late in life, of his polynomial approximation theorem; suitably generalised as the Stone-Weierstrass theorem, it became a central tool for twentieth-century analysis. Furthermore, the Weierstrass nowhere-differentiable function is the seed from which springs the entire modern theory of mathematical finance. The best students in Europe came to Berlin to attend his lectures, and his rigorous style still dominates the first analysis course at any university. His seven-volume collected works in the original German contain not only published treatises but also records of many of his famous lecture courses. Volume 2 was published in 1895.

The German mathematician Karl Weierstrass (1815-97) is generally considered to be the father of modern analysis. His clear eye for what was important is demonstrated by the publication, late in life, of his polynomial approximation theorem; suitably generalised as the Stone-Weierstrass theorem, it became a central tool for twentieth-century analysis. Furthermore, the Weierstrass nowhere-differentiable function is the seed from which springs the entire modern theory of mathematical finance. The best students in Europe came to Berlin to attend his lectures, and his rigorous style still dominates the first analysis course at any university. His seven-volume collected works in the original German contain not only published treatises but also records of many of his famous lecture courses. Edited by Johannes Knoblauch (1855-1915), Volume 3 was published in 1903.

The German mathematician Karl Weierstrass (1815-97) is generally considered to be the father of modern analysis. His clear eye for what was important is demonstrated by the publication, late in life, of his polynomial approximation theorem; suitably generalised as the Stone-Weierstrass theorem, it became a central tool for twentieth-century analysis. Furthermore, the Weierstrass nowhere-differentiable function is the seed from which springs the entire modern theory of mathematical finance. The best students in Europe came to Berlin to attend his lectures, and his rigorous style still dominates the first analysis course at any university. His seven-volume collected works in the original German contain not only published treatises but also records of many of his famous lecture courses. Edited by Johannes Knoblauch (1885-1915) and Georg Hettner (1854-1914), Volume 4 was published in 1902.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Carl Wilhelm Borchardt (1817-80), Volume 1 appeared in 1881.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Karl Weierstrass (1815-97), Volume 2 appeared in 1882.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Karl Weierstrass (1815-97), Volume 3 appeared in 1884.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Karl Weierstrass (1815-97), Volume 4 appeared in 1886.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Karl Weierstrass (1815-97), Volume 5 appeared in 1890.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Karl Weierstrass (1815-97), Volume 6 appeared in 1891.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Karl Weierstrass (1815-97), Volume 7 appeared in 1891.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804-51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802-29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891, edited chiefly by Karl Weierstrass (1815-97). Published in 1884, this supplementary volume contains Jacobi's 1842-3 lectures on dynamics as compiled by Alfred Clebsch (1833-72) in the revised second edition by Eduard Lottner (1826-87).

Following the French Revolution, the physicist and mathematician Jean Baptiste Joseph Fourier (1768-1830) taught at the Ecole Normale Superieure and later succeeded Lagrange at the Ecole Polytechnique. He was promoted to administrative positions under Napoleon, but continued to pursue his scientific interests. From 1822 until his death he served as the permanent secretary for mathematical sciences at the Academie des Sciences. Thanks to his substantial contributions to the field, Fourier's name has passed as an adjective into the mathematical vocabulary of every major language. These selected works were edited by the mathematician Jean Gaston Darboux (1842-1917) and published in two volumes between 1888 and 1890. Volume 1 is given over entirely to the immortal Theorie analytique de la chaleur (1822), from which the world learnt about the heat equation and the series which bears Fourier's name.

Following the French Revolution, the physicist and mathematician Jean Baptiste Joseph Fourier (1768-1830) taught at the Ecole Normale Superieure and later succeeded Lagrange at the Ecole Polytechnique. He was promoted to administrative positions under Napoleon, but continued to pursue his scientific interests. From 1822 until his death he served as the permanent secretary for mathematical sciences at the Academie des Sciences. These selected works were edited by the mathematician Jean Gaston Darboux (1842-1917) and published in two volumes between 1888 and 1890. Volume 2 contains several extraordinary contributions: the first paper to address the question of why the earth's surface is warm (which we now call the greenhouse effect), the first paper to address the cooling of the earth's interior (still a major research topic) and the first paper on optimisation under linear constraints, along with the results on roots of polynomials which first made Fourier's reputation.

This volume contains eighteen papers that have been collected by the Canadian Society for History and Philosophy of Mathematics. It showcases rigorously-reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics, as well as the teaching of the history of mathematics.   Some of the topics explored include Arabic editions of Euclid’s Elements from the thirteenth century and their role in the assimilation of Euclidean geometry into the Islamic intellectual tradition Portuguese sixteenth century recreational mathematics as found in the Tratado de Prática Darysmetica  A Cambridge correspondence course in arithmetic for women in England in the late nineteenth century The mathematical interests of the famous Egyptologist Thomas Eric (T. E.) Peet  The history of Zentralblatt für Mathematik and Mathematical Reviews and their role in creating a publishing infrastructure for a global mathematical literature The use of Latin squares for agricultural crop experiments at the Rothamsted Experimental Station The many contributions of women to the advancement of computing techniques at the Cavendish Laboratory at the University of Cambridge in the 1960s The volume concludes with two short plays, one set in Ancient Mesopotamia and the other in Ancient Egypt, that are well suited for use in the mathematics classroom. Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.

Carl Gustav Jacob Jacobi (1804-51) was one of the nineteenth century's greatest mathematicians, as attested by the diversity of mathematical objects named after him. His early work on number theory had already attracted the attention of Carl Friedrich Gauss, but his reputation was made by his work on elliptic functions. Elliptic integrals had been studied for a long time, but in 1827 Jacobi and Niels Henrik Abel realised independently that the correct way to view them was by means of their inverse functions - what we now call the elliptic functions. The next few years witnessed a flowering of the subject as the two mathematicians pushed ahead. Adrien-Marie Legendre, an expert on the old theory, wrote: 'I congratulate myself that I have lived long enough to witness these magnanimous conflicts between two equally strong young athletes'. This Latin work, first published in 1829, is Jacobi's pioneering account of the new theory.

The great nineteenth-century mathematician Peter Gustav Lejeune Dirichlet (1805-59) studied in Paris, coming under the influence of scholars including Fourier and Legendre. He then taught at Berlin and Goettingen universities, where he was the successor to Gauss and mentor to Riemann and Dedekind. His achievements include the first satisfactory proof of the convergence of Fourier series under appropriate conditions, and the theorem on primes in arithmetic progression which was, at the same time, the foundation of analytic number theory and one of its greatest achievements. He also did important work on Laplace's equation, the theory of series and many other topics. This two-volume collection of his works, published 1889-97, was compiled by Leopold Kronecker (1823-91). Volume 1 contains works published by Dirichlet up to 1843, together with a related 1846 essay.

The great nineteenth-century mathematician Peter Gustav Lejeune Dirichlet (1805-59) studied in Paris, coming under the influence of scholars including Fourier and Legendre. He then taught at Berlin and Goettingen universities, where he was the successor to Gauss and mentor to Riemann and Dedekind. His achievements include the first satisfactory proof of the convergence of Fourier series under appropriate conditions, and the theorem on primes in arithmetic progression which was, at the same time, the foundation of analytic number theory and one of its greatest achievements. He also did important work on Laplace's equation, the theory of series and many other topics. This two-volume collection of his works, published 1889-97, was compiled by Leopold Kronecker (1823-91). Volume 2 was completed by Lazarus Fuchs (1833-1902) and contains Dirichlet's publications from 1844 onwards, together with some unpublished papers and selected correspondence with Gauss, Alexander von Humboldt and Kronecker.

Few men in modern mathematics have had as great an impact as the Norwegian Niels Henrik Abel (1802-29), whose discoveries paved the way for several new branches of nineteenth-century mathematics. Tragically, Abel's short life was dominated by poverty and his scientific achievements were not fully recognised until after his death. This work, written by Carl Anton Bjerknes (1825-1903), was the first full biography of Abel. Originally published in 1880 and translated into French in 1885, it became a valuable resource for later Abel biographers and scholars of the history of mathematics. With insight and understanding, Bjerknes charts the progress of the talented young mathematician and gives a detailed account of Abel's work and his correspondence with other contemporary mathematicians. In particular, he examines in depth (from Abel's point of view) the dispute between Abel and his rival Jacobi relating to their discoveries of elliptic functions in the 1820s.

A comprehensive edition and commentary of a late antique codex Mathematics, Metrology, and Model Contracts is a comprehensive edition and commentary of a late antique codex. The codex contains mathematical problems, metrological tables, and model contracts. Given the nature of the contents, the format, and quality of the Greek, the editors conclude that the codex most likely belonged to a student in a school devoted to training business agents and similar professionals. The editors present here the first full scholarly edition of the text, with complete discussions of the provenance, codicology, and philology of the surviving manuscript. They also provide extensive notes and illustrations for the mathematical problems and model contracts, as well as historical commentary on what this text reveals about late antique numeracy, literacy, education, and vocational training in what we would now see as business, law, and administration. The book will be of interest to papyrologists and scholars who are interested in the history and culture of late antiquity, the history of education, literacy, the ancient economy, and the history of science and mathematics.

Instant Mathematics pulls together all the pivotal mathematical theories and discoveries into one concise volume. Each page contains a discrete 'cheat sheet', which tells you the most important facts in bite-sized chunks, meaning you can become an expert in an instant. From zero to the Riemann Hypothesis, from primes to irrational numbers, and from Pythagoras to John Nash and Roger Penrose, every key figure, theory or term is expressed in succinct and lively text and graphics. Perfect for the knowledge hungry and time poor, this collection of graphic-led lessons makes mathematics interesting and accessible. Everything you need to know - and more - is here.