The scholar and East India Company administrator Henry Thomas Colebrooke (1765-1837) brought India's rich mathematical heritage to the attention of the wider world with the publication of this book in 1817. Based on Sanskrit texts, it contains English translations of classic works by the Indian mathematicians and astronomers Brahmagupta (598-668) and Bhascara (1114-85), who were instrumental thinkers in the development of algebra. Included here are translations of chapters 12 and 18 of Brahmagupta's best-known work, Brahmasphutasiddhanta, focusing on arithmetic and algebra respectively. Also included in this book are translations of two of the greatest works by Bhascara: Lilavati, his treatise on arithmetic, and Bijaganita, on algebra. Furthermore, Colebrooke's introduction aims to position the Indian advancement of algebra in relation to its development by the Greeks and Arabs.

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 5 was published in 1915.

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 Rudolf Rothe (1873-1942), Volume 6 was published in 1915.

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 Rudolf Rothe (1873-1942), Volume 7 was published in 1927.

Originally published in 1918, this book forms part of a three-volume work created to expand upon the content of a series of lectures delivered at the University of Calcutta during the winter of 1909-10. The chief feature of all three volumes is that they deal with rectangular matrices and determinoids as distinguished from square matrices and determinants, the determinoid of a rectangular matrix being related to it in the same way as a determinant is related to a square matrix. An attempt is made to set forth a complete and consistent theory or calculus of rectangular matrices and determinoids. The second volume contains further developments of the general theory, including a discussion of matrix equations of the second degree. It also contains a large number of applications to algebra and to analytical geometry of space of two, three and n dimensions.

Mohammed ben Musa (c.780 c.850) was a Persian mathematician and astronomer. The word 'algebra' derives from his Compendious Book on Calculation by Completion and Balancing, which introduced modern algebraic methods. First published in 1831, this translation from Arabic into English was prepared by the German orientalist Friedrich August Rosen (1805 37). The key algebraic methods introduced are reduction, completion and balancing. To reduce an equation is to change an expression to a simpler form; completion is to remove a negative quantity from one side of the equation and add it to the other; and balancing is to cancel like terms on opposite sides of the equation. An account is also given of solving polynomial equations up to the second degree. Rosen's introduction and notes accompany the translation, which remains relevant in the history of mathematics.

An important mathematician and astronomer in medieval India, Bhascara Acharya (1114 85) wrote treatises on arithmetic, algebra, geometry and astronomy. He is also believed to have been head of the astronomical observatory at Ujjain, which was the leading centre of mathematical sciences in India. Forming part of his Sanskrit magnum opus Siddh nta Shiromani, the present work is his treatise on arithmetic, including coverage of geometry. It was first published in English in 1816 after being translated by the East India Company surgeon John Taylor (d.1821). Used as a textbook in India for centuries, it provides the basic mathematics needed for astronomy. Topics covered include arithmetical terms, plane geometry, solid geometry and indeterminate equations. Of enduring interest in the history of mathematics, this work also contains Bhascara's pictorial proof of Pythagoras' theorem.

Originally published in 1913, this book forms part of a three-volume work created to expand upon the content of a series of lectures delivered at the University of Calcutta during the winter of 1909-10. The chief feature of all three volumes is that they deal with rectangular matrices and determinoids as distinguished from square matrices and determinants, the determinoid of a rectangular matrix being related to it in the same way as a determinant is related to a square matrix. An attempt is made to set forth a complete and consistent theory or calculus of rectangular matrices and determinoids. The first volume contains the most fundamental portions of the theory and concludes with the solution of any system of linear algebraic equations, which is treated as a special case of the solution of a matrix equation of the first degree.

An important mathematician and astronomer in medieval India, Bhascara Acharya (1114 85) wrote treatises on arithmetic, algebra, geometry and astronomy. He is also believed to have been head of the astronomical observatory at Ujjain, which was the leading centre of mathematical sciences in India. Forming part of his Sanskrit magnum opus Siddh nta Shiromani, the present work is his treatise on algebra. It was first published in English in 1813 after being translated from a Persian text by the East India Company civil servant Edward Strachey (1774 1832). The topics covered include operations involving positive and negative numbers, surds and zero, as well as algebraic, simultaneous and indeterminate equations. Strachey also appends useful notes made by the orientalist Samuel Davis (1760 1819). Of enduring interest in the history of mathematics, this was notably the first work to acknowledge that a positive number has two square roots.

The authors' novel approach to some interesting mathematical concepts - not normally taught in other courses - places them in a historical and philosophical setting. Although primarily intended for mathematics undergraduates, the book will also appeal to students in the sciences, humanities and education with a strong interest in this subject. The first part proceeds from about 1800 BC to 1800 AD, discussing, for example, the Renaissance method for solving cubic and quartic equations and providing rigorous elementary proof that certain geometrical problems posed by the ancient Greeks cannot be solved by ruler and compass alone. The second part presents some fundamental topics of interest from the past two centuries, including proof of G del's incompleteness theorem, together with a discussion of its implications.

First published in 1961, this book provides information on the methods of treating series of observations, the field covered embraces portions of both statistics and numerical analysis. Originally intended as an introduction to the topic aimed at students and graduates in physics, the types of observation discussed reflect the standard routine work of the time in the physical sciences. The text partly reflects an aim to offer a better balance between theory and practice, reversing the tendency of books on numerical analysis to omit numerical examples illustrating the applications of the methods. This book will be of value to anyone with an interest in the theoretical development of its field.

Prior to the advent of computers, no mathematician, physicist or engineer could do without a volume of tables of logarithmic and trigonometric functions. These tables made possible certain calculations which would otherwise be impossible. Unfortunately, carelessness and lazy plagiarism meant that the tables often contained serious errors. Those prepared by Charles Hutton (1737 1823) were notable for their reliability and remained the standard for a century. Hutton had risen, by mathematical ability, hard work and some luck, from humble beginnings to become a professor of mathematics at the Royal Military Academy. His mathematical work was distinguished by utility rather than originality, but his contributions to the teaching of the subject were substantial. This seventh edition was published in 1858 with additional material by Olinthus Gregory (1774 1841). The preliminary matter will be of interest to any modern-day reader who wishes to know how calculation was done before the electronic computer.

Max Tegmark leads us on an astonishing journey through past, present and future, and through the physics, astronomy and mathematics that are the foundation of his work, most particularly his hypothesis that our physical reality is a mathematical structure and his theory of the ultimate multiverse. In a dazzling combination of both popular and groundbreaking science, he not only helps us grasp his often mind-boggling theories, but he also shares with us some of the often surprising triumphs and disappointments that have shaped his life as a scientist. Fascinating from first to last--this is a book that has already prompted the attention and admiration of some of the most prominent scientists and mathematicians.

The nineteenth century saw the paradoxes and obscurities of eighteenth-century calculus gradually replaced by the exact theorems and statements of rigorous analysis. It became clear that all analysis could be deduced from the properties of the real numbers. But what are the real numbers and why do they have the properties we claim they do? In this charming and influential book, Richard Dedekind (1831-1916), Professor at the Technische Hochschule in Braunschweig, showed how to resolve this problem starting from elementary ideas. His method of constructing the reals from the rationals (the Dedekind cut) remains central to this day and was generalised by Conway in his construction of the 'surreal numbers'. This reissue of Dedekind's 1888 classic is of the 'second, unaltered' 1893 edition.

Niels Henrik Abel (1802-29) was one of the most prominent mathematicians in the first half of the nineteenth century. His pioneering work in diverse areas such as algebra, analysis, geometry and mechanics has made the adjective 'abelian' a commonplace in mathematical writing. These collected works, first published in two volumes in 1881 after careful preparation by the mathematicians Ludwig Sylow (1832-1918) and Sophus Lie (1842-99), contain some of the pillars of mathematical history. Volume 1 includes perhaps the most famous of Abel's results, namely his proof of the 'impossibility theorem', which states that the general fifth-degree polynomial is unsolvable by algebraic means. Also included in this volume is Abel's 'Paris memoir', which contains his many fundamental results on transcendental functions - in particular on elliptic integrals, elliptic functions, and what are known today as abelian integrals.

Niels Henrik Abel (1802-29) was one of the most prominent mathematicians in the first half of the nineteenth century. His pioneering work in diverse areas such as algebra, analysis, geometry and mechanics has made the adjective 'abelian' a commonplace in mathematical writing. These collected works, first published in two volumes in 1881 after careful preparation by the mathematicians Ludwig Sylow (1832-1918) and Sophus Lie (1842-99), contain some of the pillars of mathematical history. Volume 2 contains additional articles on elliptic functions and infinite series. It also includes extracts from Abel's letters, as well as detailed notes and commentary by Sylow and Lie on Abel's pioneering work.

James Joseph Sylvester (1814-97) was an English mathematician who made key contributions to numerous areas of his field and was also of primary importance in the development of American mathematics, both as inaugural Professor of Mathematics at Johns Hopkins University and founder of the American Journal of Mathematics. Originally published in 1904, this book forms the first in four volumes of Sylvester's mathematical papers, covering the period from 1837 to 1853. Together these volumes provide a comprehensive resource that will be of value to anyone with an interest in Sylvester's theories and the history of mathematics.

James Joseph Sylvester (1814-97) was an English mathematician who made key contributions to numerous areas of his field and was also of primary importance in the development of American mathematics, both as inaugural Professor of Mathematics at Johns Hopkins University and founder of the American Journal of Mathematics. Originally published in 1908, this book forms the second in four volumes of Sylvester's mathematical papers, covering the period from 1854 to 1873. Together these volumes provide a comprehensive resource that will be of value to anyone with an interest in Sylvester's theories and the history of mathematics.

James Joseph Sylvester (1814-97) was an English mathematician who made key contributions to numerous areas of his field and was also of primary importance in the development of American mathematics, both as inaugural Professor of Mathematics at Johns Hopkins University and founder of the American Journal of Mathematics. Originally published in 1909, this book forms the third in four volumes of Sylvester's mathematical papers, covering the period from 1870 to 1883. Together these volumes provide a comprehensive resource that will be of value to anyone with an interest in Sylvester's theories and the history of mathematics.

James Joseph Sylvester (1814-97) was an English mathematician who made key contributions to numerous areas of his field and was also of primary importance in the development of American mathematics, both as inaugural Professor of Mathematics at Johns Hopkins University and founder of the American Journal of Mathematics. Originally published in 1912, this book forms the fourth in four volumes of Sylvester's mathematical papers, covering the period from 1882 to 1897. Together these volumes provide a comprehensive resource that will be of value to anyone with an interest in Sylvester's theories and the history of mathematics.

First published in 1931, this book is the second edition of a 1917 original. The text provides an account of the method of least squares, aiming to obtain the best interpretation of the results of experiment without consideration of the way in which these results are obtained. Elaborate descriptions of instruments and experimental methods are avoided, allowing for a concise and economical account that concentrates on key elements of the subject. This is a detailed and well-organized book that will be of value to anyone with an interest in least squares and the development of mathematics.

Gerade heute, wo sich die Aufmerksamkeit der fuhrenden Philosophen, Logiker und Mathematiker erneut auf die Grundlagen der systematisch-deduktiven Mathematik richtet, ist dieses Buch von zeitnaher und tiefer Bedeutung."

1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's Elemen ta, where we find a proof of their infinitude, now regarded as canonical. One feels that Euclid's argument has its place in The Book, often quoted by the late Paul ErdOs, where the ultimate forms of mathematical arguments are preserved. Proofs of most other results on prime number distribution seem to be still far away from their optimal form and the aim of this book is to present the development of methods with which such problems were attacked in the course of time. This is not a historical book since we refrain from giving biographical details of the people who have played a role in this development and we do not discuss the questions concerning why each particular person became in terested in primes, because, usually, exact answers to them are impossible to obtain. Our idea is to present the development of the theory of the distribu tion of prime numbers in the period starting in antiquity and concluding at the end of the first decade of the 20th century. We shall also present some later developments, mostly in short comments, although the reader will find certain exceptions to that rule. The period of the last 80 years was full of new ideas (we mention only the applications of trigonometrical sums or the advent of various sieve methods) and certainly demands a separate book."

The International Bureau of Weights and Measures (BIPM) is currently implementing the greatest change ever in the world's system of weights and measures -- it is redefining the kilogram, the final artefact standard, and reorganizing the system of international units. This book tells the inside story of what led to these changes, from the events surrounding the founding of the BIPM in 1875 -- a landmark in the history of international cooperation -- to the present. It traces not only the evolution of the science, but also the story of the key individuals and events.
The BIPM was the first international scientific laboratory. Founded in 1875 by the Metre Convention, its original tasks were to conserve the new international standards of the metre and the kilogram, to carry out calibrations for Member States and undertake research to advance measurement science. The book is based on the substantial archive of the BIPM which, from the very beginning, recounts the many discussions and arguments first as to whether and how such an institute should be created and in due course, how over the next one hundred and thirty years it should develop. Despite many national and personal rivalries, the institute actually created was admirably suited to its declared tasks. In the years and decades that followed, the scientific work of the small group of men who made up its first staff was of a very high order. One of the early Directors received the Nobel Prize for physics in 1920 for his discovery of invar. The international governing Board of the institute, the International Committee of Weights and Measures, has guided the institute from one charged with the conservation of the prototype artefacts to one now at the centre of world metrology and preparing for the redefinition of the last remaining artifact, the kilogram, in terms of a fixed value for one of the fundamental constants of physics, the Planck constant

At the turn of the twentieth century, mathematical scholarship in the United States underwent a stunning transformation. In 1890, no American professor was producing mathematical research worthy of international attention. Graduate students were then advised to pursue their studies abroad. By the start of World War I, the standing of American mathematics had radically changed. George David Birkhoff, Leonard Dickson, and others were turning out cutting edge investigations that attracted notice in the intellectual centers of Europe. Harvard, Chicago, and Princeton maintained graduate programs comparable to those overseas. This book explores the people, timing, and factors behind this rapid advance. Through the mid-nineteenth century, most American colleges followed a classical curriculum that, in mathematics, rarely reached beyond calculus. With no doctoral programs of any sort in the United States until 1860, mathematical scholarship lagged far behind that in Europe. After the Civil War, visionary presidents at Harvard and Johns Hopkins broadened and deepened the opportunities for study. The breakthrough for mathematics began in 1890 with the hiring, in consecutive years, of William F. Osgood and Maxime Bocher at Harvard and E. H. Moore at Chicago. Each of these young men had studied in Germany where they acquired vital mathematical knowledge and taste. Over the next few years, Osgood, Bocher, and Moore established their own research programs and introduced new graduate courses. Working with other like-minded individuals through the nascent American Mathematical Society, the infrastructure of meetings and journals were created. In the early twentieth century, Princeton dramatically upgraded its faculty to give the United States the stability of a third mathematics center. The publication by Birkhoff, in 1913, of the solution to a famous conjecture served notice that American mathematics had earned consideration with the European powers of Germany, France, Italy, England, and Russia.