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INTRODUCTION CHAPTER �� THE FIVE GROUPS OF AXIOMS ��1.The elements of geometry and the five groups of axioms ��2.Group I.Axioms of connection ��3.Group II.Axioms of order ��4.Consequences of the axioms of connection and order ��5.Group III.Axiom of parallels (Euclid's axiom) ��6.Group IV.Axioms of congruence ��7.Consequences of the axioms of congruence ��8.Group V.Axiom of continuity (Archimedes's axiom) CHAPTER �� COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS ��9.Compatibility of the axioms ��10.Independence of the axioms of parallels (Non-euclidean geom-etry) ��11.Independence of the axioms of congruence ��12.Independence of the axiom of continuity (Non-archimedean geometry) CHAPTER �� THE THEORY OF PROPORTION ��13.Complex number systems ��14.Demonstration of Pascal's theorem ��15.An algebra of segments, based upon Pascal's theorem ��16.Proportion and the theorems of similitude ��17.Equations of straight lines and of planes CHAPTER �� THE THEORY OF PLANE AREAS ��18.Equal area and equal content of polygons ��19.Parallelograms and triangles having equal bases and equal al-titudes ��20.The measure of area of triangles and polygons ��21.Equality of content and the measure of area CHAPTER �� DESARGUES'S THEOREM ��22.Desargues's theorem and its demonstration for plane geometry by aid of the axioms of congruence ��23.The impossibility of demonstrating Desargues's theorem for the plane without the help of the axioms of congruence ��24.Introduction of an algebra of segments based upon Desargues's theorem and independent of the axioms of congruence ��25.The commutative and the associative law of addition for our new algebra of segments ��26.The associative law of multiplication and the two distributive laws for the new algebra of segments ��27.Equation of the straight line, based upon the new algebra of segments ��28.The totality of segments, regarded as a complex number sys-tem ��29.Construction of a geometry of space by aid of a desarguesian number system ��30.Significance of Desargues's theorem CHAPTER �� PASCAL'S THEOREM ��31.Two theorems concerning the possibility of proving Pascal's theorem ��32.The commutative law of multiplication for an archimedean number system ��33.The commutative law of multiplication for a non-archimedean number system ��34.Proof of the two propositions concerning Pascal's theorem (Non-pascalian geometry) ��35.The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection CHAPTER �� GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I-V ��36.Geometrical constructions by means of a straight-edge and a transferer of segments ��37.Analytical representation of the co-ordinates of points which can be so constructed ��38.The representation of algebraic numbers and of integral rational functions as sums of squares ��39.Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments CONCLUSION APPENDIX