# The Geometry of Rene Descartes

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## Description

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
- 135 x 202 x 13.21mm | 276.69g
- 17 Mar 2003
- Dover Publications Inc.
- New York, United States
- English
- 0486600688
- 9780486600680
- 374,190

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## 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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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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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

show more