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From its pre-historic roots in simple counting to the algorithms powering modern desktop computers, from the genius of Archimedes to the genius of Einstein, advances in mathematical understanding and numerical techniques have been directly responsible for creating the modern world as we know it. This series will provide a library of the most influential publications and writers on mathematics in its broadest sense. As such, it will show not only the deep roots from which modern science and technology have grown, but also the astonishing breadth of application of mathematical techniques in the humanities and social sciences and in everyday life.

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182 results in Cambridge Library Collection - Mathematics

Page 1 of 10

Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix
  • Nicolas de Condorcet
  • Published online:
    05 June 2016
    Print publication:
    25 September 2014
    First published in:
    1785
  • A central figure in the early years of the French Revolution, Nicolas de Condorcet (1743–94) was active as a mathematician, philosopher, politician and economist. He argued for the values of the Enlightenment, from religious toleration to the abolition of slavery, believing that society could be improved by the application of rational thought. In this essay, first published in 1785, Condorcet analyses mathematically the process of making majority decisions, and seeks methods to improve the likelihood of their success. The work was largely forgotten in the nineteenth century, while those who did comment on it tended to find the arguments obscure. In the second half of the twentieth century, however, it was rediscovered as a foundational work in the theory of voting and societal preferences. Condorcet presents several significant results, among which Condorcet's paradox (the non-transitivity of majority preferences) is now seen as the direct ancestor of Arrow's paradox.

A History of the Mathematical Theory of Probability
  • From the Time of Pascal to that of Laplace
  • Isaac Todhunter
  • Published online:
    05 June 2016
    Print publication:
    25 September 2014
    First published in:
    1865
  • 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 Méré, 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.

The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies
  • Edward John Routh
  • Published online:
    05 December 2015
    Print publication:
    15 November 2012
    First published in:
    1891
  • As senior wrangler in 1854, Edward John Routh (1831–1907) was the man who beat James Clerk Maxwell in the Cambridge mathematics tripos. He went on to become a highly successful coach in mathematics at Cambridge, producing a total of twenty-seven senior wranglers during his career - an unrivalled achievement. In addition to his considerable teaching commitments, Routh was also a very able and productive researcher who contributed to the foundations of control theory and to the modern treatment of mechanics. First published in one volume in 1860, this textbook helped disseminate Routh's investigations into stability. This revised fifth edition was published in two volumes between 1891 and 1892. The first part establishes the principles of dynamics, providing formulae and examples throughout. While the growth of modern physics and mathematics may have forced out the problem-based mechanics of Routh's textbooks from the undergraduate syllabus, the utility and importance of his work is undiminished.

Was sind und was sollen die Zahlen?
  • Richard Dedekind
  • Published online:
    05 November 2015
    Print publication:
    02 August 2012
    First published in:
    1893
  • 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.

A Budget of Paradoxes
  • Augustus De Morgan
  • Edited by Sophia De Morgan
  • Published online:
    05 September 2015
    Print publication:
    30 April 2015
    First published in:
    1872
  • An important figure in the development of modern mathematical logic and abstract algebra, Augustus De Morgan (1806–71) was also a witty writer who made a hobby of collecting evidence of paradoxical and illogical thinking from historical sources as well as contemporary pamphlets and periodicals. Based on articles that had appeared in The Athenaeum during his lifetime, this work was edited by his widow and published in book form in 1872. It parades all varieties of crackpot, from circle-squarers to inventors of perpetual motion machines, all for the reader's entertainment and education. Filled with anecdotes, personal opinions and 'squibs' of every kind, the book remains enjoyable reading for those who are amused rather than appalled by the human condition. Also reissued in the Cambridge Library Collection are the Memoir of Augustus De Morgan (1882), prepared by his wife, and his ambitious Formal Logic (1847).

Elements of Quaternions
  • William Rowan Hamilton
  • Edited by William Edwin Hamilton
  • Published online:
    05 July 2015
    Print publication:
    10 June 2010
    First published in:
    1866
  • Sir William Rowan Hamilton (1805–65) was a distinguished Irish mathematician who worked in the fields of classical mechanics, optics and algebra, as well as in physics and astronomy. Hamilton was the discoverer of quaternions, which are defined as a non-commutative number system which extends the complex numbers. He first described them in 1843, and devoted much of his subsequent life to studying and lecturing on the concept. This book was published posthumously in 1866, with the final editing by his son. Until they were replaced, from the mid–1880s, by vector analysis, quaternions were taught as a major topic in advanced mathematics at most universities, and their utility in describing spatial relations has led to a revival of interest in them since the late twentieth century.

Oeuvres complètes
  • Series 1
  • Volume 1
  • Augustin-Louis Cauchy
  • Published online:
    05 June 2015
    Print publication:
    20 July 2009
    First published in:
    1882
  • Augustin-Louis, Baron Cauchy (1789–1857) was the pre-eminent French mathematician of the nineteenth century. He began his career as a military engineer during the Napoleonic Wars, but even then was publishing significant mathematical papers, and was persuaded by Lagrange and Laplace to devote himself entirely to mathematics. His greatest contributions are considered to be the Cours d'analyse de l'École Royale Polytechnique (1821), Résumé des leçons sur le calcul infinitésimal (1823) and Leçons sur les applications du calcul infinitésimal à la géométrie (1826–8), and his pioneering work encompassed a huge range of topics, most significantly real analysis, the theory of functions of a complex variable, and theoretical mechanics. Twenty-six volumes of his collected papers were published between 1882 and 1958. The first series (volumes 1-12) consists of papers published by the Académie des Sciences de l'Institut de France; the second series (volumes 13-26) of papers published elsewhere.

Formal Logic
  • Or, The Calculus of Inference, Necessary and Probable
  • Augustus De Morgan
  • Published online:
    05 June 2015
    Print publication:
    21 August 2014
    First published in:
    1847
  • 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.

Mécanique Analytique
  • Volume 1
  • Joseph-Louis Lagrange
  • Published online:
    05 February 2015
    Print publication:
    20 July 2009
    First published in:
    1788
  • Joseph-Louis Lagrange (1736–1813), one of the notable French mathematicians of the Revolutionary period, is remembered for his work in the fields of analysis, number theory and mechanics. Like Laplace and Legendre, Lagrange was assisted by d'Alembert, and it was on the recommendation of the latter and the urging of Frederick the Great himself that Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin. The two-volume Mécanique analytique was first published in 1788; the edition presented here is that of 1811–15, revised by the author before his death. In this work, claimed to be the most important on classical mechanics since Newton, Lagrange developed the law of virtual work, from which single principle the whole of solid and fluid mechanics can be derived.

Arithmetical Books from the Invention of Printing to the Present Time
  • Being Brief Notices of a Large Number of Works Drawn Up from Actual Inspection
  • Augustus De Morgan
  • Published online:
    18 December 2014
    Print publication:
    21 August 2014
    First published in:
    1847
  • In the preface to this work, mathematician Augustus De Morgan (1806–71) claims that 'The most worthless book of a bygone day is a record worthy of preservation.' His purpose in writing this catalogue, published in 1847, was to provide an accurate record of the early history of publishing on arithmetic, but describing only those books which he had examined himself. He surveyed the library of the Royal Society, works in the British Museum, the wares of specialist booksellers, and the private collections of himself and his friends to compile a chronological list of books from 1491 to 1846 (the final book being a work of his own), giving bibliographical details, a description of the contents, and sometimes comments on the mathematics on display. De Morgan's Formal Logic and a Memoir of Augustus De Morgan by his widow are also reissued in the Cambridge Library Collection.

A History of the Theory of Elasticity and of the Strength of Materials
  • From Galilei to the Present Time
  • Volume 2, Saint-Venant to Lord Kelvin, Part 2
  • Isaac Todhunter
  • Edited by Karl Pearson
  • Published online:
    05 December 2014
    Print publication:
    20 March 2014
    First published in:
    1893
  • 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 History of the Theory of Elasticity and of the Strength of Materials
  • From Galilei to the Present Time
  • Volume 2, Saint-Venant to Lord Kelvin, Part 1
  • Isaac Todhunter
  • Edited by Karl Pearson
  • Published online:
    05 December 2014
    Print publication:
    20 March 2014
    First published in:
    1893
  • 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.

C. G. J. Jacobi's Gesammelte Werke
  • Herausgegeben auf Veranlassung der königlich preussischen Akademie der Wissenschaften
  • Volume 6
  • Carl Gustav Jacob Jacobi
  • Edited by Karl Weierstrass
  • Published online:
    05 December 2014
    Print publication:
    18 April 2013
    First published 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 6 appeared in 1891.

A Treatise on Differential Equations
  • George Boole
  • Published online:
    05 December 2014
    Print publication:
    21 August 2014
  • The need to support his family meant that George Boole (1815–64) was a largely self-educated mathematician. Widely recognised for his ability, he became the first professor of mathematics at Cork. Boole belonged to the British school of algebra, which held what now seems to modern mathematicians to be an excessive belief in the power of symbolism. However, in Boole's hands symbolic algebra became a source of novel and lasting mathematics. Also reissued in this series, his masterpiece was An Investigation of the Laws of Thought (1854), and his two later works A Treatise on Differential Equations (1859) and A Treatise on the Calculus of Finite Differences (1860) exercised an influence which can still be traced in many modern treatments of differential equations and numerical analysis. The beautiful and mysterious formulae that Boole obtained are among the direct ancestors of the theories of distributions and of operator algebras.

C. G. J. Jacobi's Gesammelte Werke
  • Herausgegeben auf Veranlassung der königlich preussischen Akademie der Wissenschaften
  • Volume 7
  • Carl Gustav Jacob Jacobi
  • Edited by Karl Weierstrass
  • Published online:
    05 December 2014
    Print publication:
    18 April 2013
    First published 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.

Tables Requisite to Be Used with the Nautical Ephemeris, for Finding the Latitude and Longitude at Sea
  • Published by Order of the Commissioners of Longitude
  • Nevil Maskelyne
  • Published online:
    05 November 2014
    Print publication:
    20 March 2014
  • 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.

The Elements of Algebra
  • Designed for the Use of Students in the University
  • James Wood
  • Published online:
    05 November 2014
    Print publication:
    02 January 2014
  • 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.

A History of the Theory of Elasticity and of the Strength of Materials
  • From Galilei to the Present Time
  • Volume 1, From Galilei to Saint-Venant
  • Isaac Todhunter
  • Edited by Karl Pearson
  • Published online:
    05 November 2014
    Print publication:
    20 March 2014
    First published in:
    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.

Cours d'analyse de l'ecole polytechnique
  • Volume 1, Calcul différentiel
  • Camille Jordan
  • Published online:
    05 October 2014
    Print publication:
    21 November 2013
  • One of the great algebraists of the nineteenth century, Marie Ennemond Camille Jordan (1838–1922) became known for his work on matrices, Galois theory and group theory. However, his most profound effect on how we see mathematics came through his Cours d'analyse, which appeared in three editions. Reissued here is the first edition, which was published in three volumes between 1882 and 1887. While highly influential in its time, it now appears to us a transitional work between the partially rigorous 'epsilon delta' calculus of Cauchy and his successors, and the new 'real number' analysis of Weierstrass and Cantor. The first two volumes follow the old tradition while the third volume incorporates a substantial amount of the new analysis. Ten years later, the even more influential second edition followed the new point of view from its start. Volume 1 (1882) covers differential calculus.

Mathematische Werke
  • Herausgegeben unter Mitwirkung einer von der königlich preussischen Akademie der Wissenschaften eingesetzten Commission
  • Volume 1
  • Karl Weierstrass
  • Published online:
    05 October 2014
    Print publication:
    18 April 2013
    First 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 1 was published in 1894.

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