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The Geometry of René Descartes

by

The Geometry of René Descartes Cover

 

Synopses & Reviews

Publisher Comments:

This is an unabridged reprint of the definitive English translations 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."

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; On the Construction of Solid and Supersolid Problems. Interweaved 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.

Synopsis:

The great work that founded analytical geometry. Includes the original French text, Descartes' own diagrams, and the definitive Smith-Latham translation. "The greatest single step ever made in the progress of the exact sciences." — John Stuart Mill.

Synopsis:

"The best book available for non-mathematicians." — Contemporary Psychology.

This book represents the earliest clear, detailed, precise exposition of the central ideas and results of game theory and related decision-making models — unencumbered by technical mathematical details. It offers a comprehensive, time-tested conceptual introduction, with a social science orientation, to a complex of ideas related to game theory including decision theory, modern utility theory, the theory of statistical decisions, and the theory of social welfare functions.

The first three chapters provide a general introduction to the theory of games including utility theory. Chapter 4 treats two-person, zero-sum games. Chapters 5 and 6 treat two-person, nonzero-sum games and concepts developed in an attempt to meet some of the deficiencies in the von Neumann-Morgenstern theory. Chapters 7-12 treat n-person games beginning with the von Neumann-Morgenstern theory and reaching into many newer developments. The last two chapters, 13 and 14, discuss individual and group decision making. Eight helpful appendixes present proofs of the famous minimax theorem, several geometric interpretations of two-person zero-sum games, solution procedures, infinite games, sequential compounding of games, and linear programming.

Thought-provoking and clearly expressed, Games and Decisions: Introduction and Critical Survey is designed for the non-mathematician and requires no advanced mathematical training. It will be welcomed by economists concerned with economic theory, political scientists and sociologists dealing with conflict of interest, experimental psychologists studying decision making, management scientists, philosophers, statisticians, and a wide range of other decision-makers. It will likewise be indispensable for students in courses in the mathematical theory of games and linear programming.

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

Product Details

ISBN:
9780486600680
Translator:
David Eugene Smith
Translator:
Marcia L. Latham
Author:
Mathematics
Author:
Descartes, Ren
Author:
eacute
Author:
Descartes, Rene
Author:
&
Author:
Descartes
Author:
Ren
Publisher:
Dover Publications
Subject:
General
Subject:
Descartes, rene, 1596-1650
Subject:
Geometry - General
Subject:
Geometry
Subject:
Geometry, analytic
Subject:
Early works to 1800
Subject:
General Mathematics
Subject:
Mathematics-Geometry - General
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
19540631
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
244
Dimensions:
8.5 x 5.38 in 0.63 lb

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The Geometry of René Descartes New Trade Paper
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Product details 244 pages Dover Publications - English 9780486600680 Reviews:
"Synopsis" by ,
The great work that founded analytical geometry. Includes the original French text, Descartes' own diagrams, and the definitive Smith-Latham translation. "The greatest single step ever made in the progress of the exact sciences." — John Stuart Mill.
"Synopsis" by ,
"The best book available for non-mathematicians." — Contemporary Psychology.

This book represents the earliest clear, detailed, precise exposition of the central ideas and results of game theory and related decision-making models — unencumbered by technical mathematical details. It offers a comprehensive, time-tested conceptual introduction, with a social science orientation, to a complex of ideas related to game theory including decision theory, modern utility theory, the theory of statistical decisions, and the theory of social welfare functions.

The first three chapters provide a general introduction to the theory of games including utility theory. Chapter 4 treats two-person, zero-sum games. Chapters 5 and 6 treat two-person, nonzero-sum games and concepts developed in an attempt to meet some of the deficiencies in the von Neumann-Morgenstern theory. Chapters 7-12 treat n-person games beginning with the von Neumann-Morgenstern theory and reaching into many newer developments. The last two chapters, 13 and 14, discuss individual and group decision making. Eight helpful appendixes present proofs of the famous minimax theorem, several geometric interpretations of two-person zero-sum games, solution procedures, infinite games, sequential compounding of games, and linear programming.

Thought-provoking and clearly expressed, Games and Decisions: Introduction and Critical Survey is designed for the non-mathematician and requires no advanced mathematical training. It will be welcomed by economists concerned with economic theory, political scientists and sociologists dealing with conflict of interest, experimental psychologists studying decision making, management scientists, philosophers, statisticians, and a wide range of other decision-makers. It will likewise be indispensable for students in courses in the mathematical theory of games and linear programming.

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