The Geometry of Rene Descartes
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The Geometry of Rene Descartes : With a Facsimile of the First Edition

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This is an unabridged republication of the definitive English translation of one of the very greatest classics of science. Originally published in 1637, it has been characterized as "the greatest single step ever made in the progress of the exact sciences" (John Stuart Mill); as a book which "remade geometry and made modern geometry possible" (Eric Temple Bell). It "revolutionized the entire conception of the object of mathematical science" (J. Hadamard).With this volume Descartes founded modern analytical geometry. Reducing geometry to algebra and analysis and, conversely, showing that analysis may be translated into geometry, it opened the way for modern mathematics. Descartes was the first to classify curves systematically and to demonstrate algebraic solution of geometric curves. His geometric interpretation of negative quantities led to later concepts of continuity and the theory of function. The third book contains important contributions to the theory of equations.This edition contains the entire definitive Smith-Latham translation of Descartes' three books: Problems the Construction of which Requires Only Straight Lines and Circles; On the Nature of Curved Lines; and On the Construction of Solid and Supersolid Problems. Interleaved page by page with the translation is a complete facsimile of the 1637 French text, together with all Descartes' original illustrations; 248 footnotes explain the text and add further bibliography.
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Product details

  • Paperback | 244 pages
  • 137.16 x 198.12 x 15.24mm | 249.47g
  • New York, United States
  • English
  • 0486600688
  • 9780486600680
  • 316,676

Back cover copy

This is an unabridged republication of the definitive English translation of one of the very greatest classics of science. Originally published in 1637, it has been characterized as "the greatest single step ever made in the progress of the exact sciences" (John Stuart Mill); as a book which "remade geometry and made modern geometry possible" (Eric Temple Bell). It "revolutionized the entire conception of the object of mathematical science" (J. Hadamard).With this volume Descartes founded modern analytical geometry. Reducing geometry to algebra and analysis and, conversely, showing that analysis may be translated into geometry, it opened the way for modern mathematics. Descartes was the first to classify curves systematically and to demonstrate algebraic solution of geometric curves. His geometric interpretation of negative quantities led to later concepts of continuity and the theory of function. The third book contains important contributions to the theory of equations.This edition contains the entire definitive Smith-Latham translation of Descartes' three books: Problems the Construction of which Requires Only Straight Lines and Circles; On the Nature of Curved Lines; and On the Construction of Solid and Supersolid Problems. Interleaved page by page with the translation is a complete facsimile of the 1637 French text, together with all Descartes' original illustrations; 248 footnotes explain the text and add further bibliography.
Translated by David E. Smith and Marcia L. Latham.
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Table of contents

BOOK I
  PROBLEMS THE CONSTRUCTION OF WHICH REQUIRES ONLY STRAIGHT LINES AND CIRCLES
    How the calculations of arithmetic are related to the operations of geometry
    "How the multiplication, division, and the extraction of square root are performed geometrically"
    How we use arithmetic symbols in geometry
    How we use equations in solving problems
    Plane problems and their solution
    Example from Pappus
    Solution of the problem of Pappus
    How we should choose the terms in arriving at the equation in this case
    How we find that this problem is plane when not more than five lines are given
BOOK II
  ON THE NATURE OF CURVED LINES
    What curved lines are admitted in geometry
    "The method of distinguishing all curved lines of certain classes, and of knowing the ratios connecting their points on certain straight lines"
    There follows the explanation of the problem of Pappus mentioned in the preceding book
    Solution of this problem for the case of only three or four lines
    Demonstration of this solution
    Plane and solid loci and the method of finding them
    The first and simplest of all the curves needed in solving the ancient problem for the case of five lines
    Geometric curves that can be described by finding a number of their points
    Those which can be described with a string
    "To find the properties of curves it is necessary to know the relation of their points to points on certain straight lines, and the method of drawing other lines which cut them in all these points at right angles"
    General method for finding straight lines which cut given curves and make right angles with them
    Example of this operation in the case of an ellipse and of a parabola of the second class
    Another example in the case of an oval of the second class
    Example of the construction of this problem in the case of the conchoid
    Explanation of four new classes of ovals which enter into optics
    The properties of these ovals relating to reflection and refraction
    Demonstration of these properties
    "How it is possible to make a lens as convex or concave as we wish, in one of its surfaces, which shall cause to converge in a given point all the rays which proceed from another given point"
    How it is possible to make a lens which operates like the preceeding and such that the convexity of one of its surfaces shall have a given ratio to the convexity or concavity of the other
    "How it is possible to apply what has been said here concerning curved lines described on a plane surface to those which are described in a space of three dimensions, or on a curved surface"
BOOK III
  ON THE CONSTRUCTION OF SOLID OR SUPERSOLID PROBLEMS
    On those curves which can be used in the construction of every problem
    Example relating to the finding of several mean proportionals
    On the nature of equations
    How many roots each equation can have
    What are false roots
    How it is possible to lower the degree of an equation when one of the roots is known
    How to determine if any given quantity is a root
    How many true roots an equation may have
    "How the false roots may become true, and the true roots false"
    How to increase or decrease the roots of an equation
    "That by increasing the true roots we decrease the false ones, and vice versa"
    How to remove the second term of an equation
    How to make the false roots true without making the true ones false
    How to fill all the places of an equation
    How to multiply or divide the roots of an equation
    How to eliminate the fractions in an equation
    How to make the known quantity of any term of an equation equal to any given quantity
    That both the true and the false roots may be real or imaginary
    The reduction of the cubic equations when the problem is plane
    The method of dividing an equation by a binomial which contains a root
    Problems which are solid when the equation is cubic
    The reduction of equations of the fourth degree when the problem is plane
      Solid problems
    Example showing the use of these reductions
    General rule for reducing equations about the fourth degree
    General method for constructing all solid problems which reduce to an equation of the third or the fourth degree
    The finding of two mean proportionals
    The trisection of an angle
    That all solid problems can be reduced to these two constructions
    The method of expressing all the roots of cubic equations and hence of all equations extending to the fourth degree
    "Why solid problems cannot be constructed without conic sections, nor those problems which are more complex without other lines that are also more complex"
    General method for constructing all problems which require equations of degree not higher than the sixth
    The finding of four mean proportionals
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141 ratings
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3 18% (26)
2 6% (9)
1 1% (1)
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