# Book:René Descartes/La Géométrie

## René Descartes: La Géométrie

Published $\text {1637}$.

### 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