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The Geometry of Rene Descartes

The Geometry of Rene Descartes

Paperback Dover Books on Mathematics Language: English / French / Latin

By (author) Rene Descartes

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  • Publisher: Dover Publications Inc.
  • Format: Paperback | 244 pages
  • Language: English / French / Latin
  • Dimensions: 137mm x 198mm x 15mm | 249g
  • Publication date: 7 August 1999
  • Publication City/Country: New York
  • ISBN 10: 0486600688
  • ISBN 13: 9780486600680
  • Sales rank: 204,939

Product description

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

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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