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
297
How multiplication, division, and the extraction of square root are performed geometrically
293
How we use arithmetic symbols in geometry
299
How we use equations in solving problems
300
Plane problems and their solution
302
Example from Pappus
304
Solution of the problem of Pappus
307
How we should choose the terms in arriving at the equation in this case
310
How we find that this problem is plane when not more than five lines are given
313
Book II
On the Nature of Curved Lines
What curved lines are admitted in geometry
315
The method of distinguishing all curved lines of certain classes, and of knowing the fatios connecting their points on certain straight lines
319
There follows the explanation of the problem of Pappus mentioned in the preceding book
323
Solution of this problem for the case of only three or four lines
324
Demonstration of this solution
332
Plane and solid loci and the method of finding them
334
The first and simplest of all the curves needed in solving the ancient problem for the case of five lines
335
Geometric curves that can be described by finding a number of their points
340
Those which can be described with a string
340
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
341
General method for finding straight lines which cut given curves and make right angles with them
342
Example of this operation in the case of an ellipse and of a parabola of the second class
343
Another example in the case of an oval of the second class
344
Example of the construction of this problem in the case of the conchoid
351
Explanation of four new classes of ovals which enter into optics
352
The properties of these ovals relating to reflection and refraction
357
Demonstration of these properties
360
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
363
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
366
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
368
Book III
On the Construction of Solid or Supersolid Problems
On those curves which can be used in the construction of every problem
369
Example relating to the finding of several mean proportionals
370
On the nature of equations
371
How many roots each equation can have
372
What are false roots
372
How it is possible to lower the degree of an equation when one of the roots is known
372
How to determine if any given quantity is a root
373
How many true roots an equation may have
373
How the false roots may become true, and the true roots false
373
How to increase or decrease the roots of an equation
374
That by increasing the true roots we decrease the false ones, and vice versa
375
How to remove the second term of an equation
376
How to make the false roots true without making the true ones false
377
How to fill all the places of an equation
378
How to multiply or divide the roots of an equation
379
How to eliminate the fractions in an equation
379
How to make the known quantity of any term of an equation equal to any given quantity
380
That both the true and the false roots may be real or imaginary
380
The reduction of cubic equations when the problem is plane
380
The method of dividing an equation by a binomial which contains a root
381
Problems which are solid when the equation is cubic
383
The reduction of equations of the fourth degree when the problem is plane, Solid problems
383
Example showing the use of these reductions
387
General rule for reducing equations above the fourth degree
389
General method for constructing all solid problems which reduce to an equation of the third or the fourth degree
389
The finding of two mean proportionals
395
The trisection of an angle
396
That all solid problems can be reduced to these two constructions
397
The method of expressing all the roots of cubic equations and hence of all equations extending to the fourth degree
400
Why solid problems cannot be constructed without conic sections, nor those problems which are more complex without other lines that are also more complex
401
General method for constructing all problems which require equations of degree not higher than the sixth
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