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INTRODUCTION CHAPTER ¢ñ THE FIVE GROUPS OF AXIOMS ¡ì1.The elements of geometry and the five groups of axioms ¡ì2.Group I.Axioms of connection ¡ì3.Group II.Axioms of order ¡ì4.Consequences of the axioms of connection and order ¡ì5.Group III.Axiom of parallels (Euclid's axiom) ¡ì6.Group IV.Axioms of congruence ¡ì7.Consequences of the axioms of congruence ¡ì8.Group V.Axiom of continuity (Archimedes's axiom) CHAPTER ¢ò COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS ¡ì9.Compatibility of the axioms ¡ì10.Independence of the axioms of parallels (Non-euclidean geom-etry) ¡ì11.Independence of the axioms of congruence ¡ì12.Independence of the axiom of continuity (Non-archimedean geometry) CHAPTER ¢ó THE THEORY OF PROPORTION ¡ì13.Complex number systems ¡ì14.Demonstration of Pascal's theorem ¡ì15.An algebra of segments, based upon Pascal's theorem ¡ì16.Proportion and the theorems of similitude ¡ì17.Equations of straight lines and of planes CHAPTER ¢ô THE THEORY OF PLANE AREAS ¡ì18.Equal area and equal content of polygons ¡ì19.Parallelograms and triangles having equal bases and equal al-titudes ¡ì20.The measure of area of triangles and polygons ¡ì21.Equality of content and the measure of area CHAPTER ¢õ DESARGUES'S THEOREM ¡ì22.Desargues's theorem and its demonstration for plane geometry by aid of the axioms of congruence ¡ì23.The impossibility of demonstrating Desargues's theorem for the plane without the help of the axioms of congruence ¡ì24.Introduction of an algebra of segments based upon Desargues's theorem and independent of the axioms of congruence ¡ì25.The commutative and the associative law of addition for our new algebra of segments ¡ì26.The associative law of multiplication and the two distributive laws for the new algebra of segments ¡ì27.Equation of the straight line, based upon the new algebra of segments ¡ì28.The totality of segments, regarded as a complex number sys-tem ¡ì29.Construction of a geometry of space by aid of a desarguesian number system ¡ì30.Significance of Desargues's theorem CHAPTER ¢ö PASCAL'S THEOREM ¡ì31.Two theorems concerning the possibility of proving Pascal's theorem ¡ì32.The commutative law of multiplication for an archimedean number system ¡ì33.The commutative law of multiplication for a non-archimedean number system ¡ì34.Proof of the two propositions concerning Pascal's theorem (Non-pascalian geometry) ¡ì35.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 ¡ì36.Geometrical constructions by means of a straight-edge and a transferer of segments ¡ì37.Analytical representation of the co-ordinates of points which can be so constructed ¡ì38.The representation of algebraic numbers and of integral rational functions as sums of squares ¡ì39.Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments CONCLUSION APPENDIX