# Book:René Descartes/La Géométrie

## Contents

## René Descartes: *La Géométrie*

Published $1637$.

### Subject Matter

### Contents

- Preface

- 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 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 on a given point all the rays which proceed from another given point
- How it is possible to make a lens which operates like the preceding 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 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 planet. Solid problems
- Example showing the use of these reductions
- General rule for reducing equations above 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

### Notable Translations and Editions

- 1925:
*The Geometry of René Descartes with a facsimile of the first edition*David Eugene Smith and Marcia L. Latham, Dover Publications, Inc: ISBN 0-486-60068-8

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.11$: Descartes ($1596$ – $1650$) - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): $\S 2.1$: Functions