The Geometry of Rene Descartes

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This is a new release of the original 1952 edition.
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The Geometry of René Descartes

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Overview

This is a new release of the original 1952 edition.
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Product Details

  • ISBN-13: 9781494064655
  • Publisher: Literary Licensing LLC
  • Publication date: 10/27/2013
  • Pages: 264
  • Product dimensions: 6.00 (w) x 9.00 (h) x 0.55 (d)

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 402
The finding of four mean proportionals 411
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