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The Geometry of RenÃ© Descartesby RenÃ© Descartes
Synopses & Reviews
This is an unabridged reprint of the definitive English translations 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."
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; On the Construction of Solid and Supersolid Problems. Interweaved 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.
The great work that founded analytical geometry. Includes the original French text, Descartes' own diagrams, and the definitive Smith-Latham translation. "The greatest single step ever made in the progress of the exact sciences." — John Stuart Mill.
"The best book available for non-mathematicians." — Contemporary Psychology.
This book represents the earliest clear, detailed, precise exposition of the central ideas and results of game theory and related decision-making models — unencumbered by technical mathematical details. It offers a comprehensive, time-tested conceptual introduction, with a social science orientation, to a complex of ideas related to game theory including decision theory, modern utility theory, the theory of statistical decisions, and the theory of social welfare functions.
The first three chapters provide a general introduction to the theory of games including utility theory. Chapter 4 treats two-person, zero-sum games. Chapters 5 and 6 treat two-person, nonzero-sum games and concepts developed in an attempt to meet some of the deficiencies in the von Neumann-Morgenstern theory. Chapters 7-12 treat n-person games beginning with the von Neumann-Morgenstern theory and reaching into many newer developments. The last two chapters, 13 and 14, discuss individual and group decision making. Eight helpful appendixes present proofs of the famous minimax theorem, several geometric interpretations of two-person zero-sum games, solution procedures, infinite games, sequential compounding of games, and linear programming.
Thought-provoking and clearly expressed, Games and Decisions: Introduction and Critical Survey is designed for the non-mathematician and requires no advanced mathematical training. It will be welcomed by economists concerned with economic theory, political scientists and sociologists dealing with conflict of interest, experimental psychologists studying decision making, management scientists, philosophers, statisticians, and a wide range of other decision-makers. It will likewise be indispensable for students in courses in the mathematical theory of games and linear programming.
Table of Contents
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
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"
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
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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