Die Werke von Jakob Bernoulli: Bd. 5: Differentialgeometrie: v. 5 - Hardcover

Bernoulli, Jakob

9783764357795: Die Werke von Jakob Bernoulli: Bd. 5: Differentialgeometrie: v. 5

Hardcover

ISBN 10: 3764357797 ISBN 13: 9783764357795

Verlag: Birkhauser Verlag AG, 1999

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This volume contains the work of the great Swiss mathematician on differential geometry, a field marked by some of his greatest achievements. Between 1690 and 1700, Jacob Bernoulli published twelve treatises in the scientific journal Acta Eruditorum on the use of infinitesimal methods to answer geometrical questions. Preparatory notes for most of these papers and on many other themes are found in Bernoulli's scientific diary Meditationes, from which twentynine texts are published here for the first time. Among the curves considered are the isochrones (lines of constant descent), the parabolic spiral, the loxodrome, the cycloid, the tractrix, and the logarithmic spiral (Bernoulli's spira mirabilis, which also adorns his tombstone). The description of these curves by differential equations and by geometrical constructions, their rectification and quadrature, and the determination of their evolutes and caustics offered Bernoulli and his colleagues a range of challenging problems, many ofthem relevant for mechanical or optical applications. The French mathematician Andr� Weil, who lived in the United States until his recent death, has greatly influenced 20th century mathematics, among other things, as a founding member of the Bourbaki group. For many years he has pursued intensive studies of the history of mathematics, especially number theory and algebraic geometry. Weil's introduction to this volume places Jacob Bernoulli's contribution to differential geometry in a line of development from Descartes, Huygens and Barrow through Newton's und Leibniz's epochal innovations right up to the codification of the subject by Euler. Martin Mattm�ller, secretary of the Bernoulli Edition at Basel, edited the source text. His commentaries consider particular topics in differential geometry with reference to their historical context at the end of the 17th century.

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Rese�a del editor

This volume contains the work of the great Swiss mathematician on differential geometry, a field marked by some of his greatest achievements. Between 1690 and 1700, Jacob Bernoulli published twelve treatises in the scientific journal Acta Eruditorum on the use of infinitesimal methods to answer geometrical questions. Preparatory notes for most of these papers and on many other themes are found in Bernoulli's scientific diary Meditationes, from which twentynine texts are published here for the first time. Among the curves considered are the isochrones (lines of constant descent), the parabolic spiral, the loxodrome, the cycloid, the tractrix, and the logarithmic spiral (Bernoulli's spira mirabilis, which also adorns his tombstone). The description of these curves by differential equations and by geometrical constructions, their rectification and quadrature, and the determination of their evolutes and caustics offered Bernoulli and his colleagues a range of challenging problems, many of them relevant for mechanical or optical applications. The French mathematician Andr� Weil, who lived in the United States until his recent death, has greatly influenced 20th century mathematics, among other things, as a founding member of the Bourbaki group. For many years he has pursued intensive studies of the history of mathematics, especially number theory and algebraic geometry. Weil's introduction to this volume places Jacob Bernoulli's contribution to differential geometry in a line of development from Descartes, Huygens and Barrow through Newton's und Leibniz's epochal innovations right up to the codification of the subject by Euler. Martin Mattm�ller, secretary of the Bernoulli Edition at Basel, edited the source text. His commentaries consider particular topics in differential geometry with reference to their historical context at the end of the 17th century.

Rese�a del editor

This volume contains the work of the great Swiss mathematician on differential geometry, a field marked by some of his greatest achievements. In the 1680s, Leibniz had given the first public hints of his "new method", differential and integral calculus. Together with Huygens, Tschirnhaus and L'Hopital, Jacob Bernoulli and his younger brother Johann belonged to the small group of scientists who developed these cryptic ideas and made use of them in their research. In the intense, sometimes competitive intellectual exchange that followed in their publications and correspondence, Leibnizian infinitesimal calculus evolved into the analytical method that was to dominate mathematical thinking for two centuries. Between 1690 and 1700, Jacob Bernoulli published 12 treatises in the scientific journal "Acta Eruditorum" on the use of infinitesimal methods to answer geometrical questions. Preparatory notes for most of these papers and on many other themes are found in Bernoulli's scientific diary "Meditationes", from which 29 texts are published here for the first time. Among the curves considered are the isochrones (lines of constant descent), the parabolic spiral, the loxodrome, the cycloid, the tractrix and the logarithmic spiral (Bernoulli's "spira mirabilis", which also adorns his tombstone). The description of these curves by differential equations and by geometrical constructions, their rectification and quadrature, and the determination of the evolutes and caustics offered Bernoulli and his colleagues a range of challenging problems, many of them relevant for mechanical or optical applications. At the same time, questions of a more general nature that were to shape the development of calculus in the 18th century gradually came into focus: the local geometry of general planar curves, their curvature and singularities; elliptic integrals and their transformation properties; and the search for algorithms for integrating algebraic functions. Through mutual suggestions, challenges and critiques, the small group of the first champions of infinitesimal calculus developed the techniques that shape differential geometry to this day. The volume concludes with Bernoulli's previously unpublished work on the classification of planar cubic curves as presented by his students Jacob Hermann and Nicolaus I Bernoulli. Though it was surpassed by Newton's contemporary research, the background of Bernoulli's contribution is of no small interest.

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