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The Foundations of Geometry电子书

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作       者:David Hilbert

出版时间:2018-02-27

字       数:16.4万

所属分类: 进口书 > 外文原版书 > 科技/医学

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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book.
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PREFACE.

INTRODUCTION.

CHAPTER I. 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. (ARCHIMEDEAN AXIOM.)

CHAPTER II. COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.

§ 9. COMPATIBILITY OF THE AXIOMS.

§ 10. INDEPENDENCE OF THE AXIOMS OF PARALLELS. (NON-EUCLIDEAN GEOMETRY.)

§ 11. INDEPENDENCE OF THE AXIOMS OF CONGRUENCE.

§ 12. INDEPENDENCE OF THE AXIOM OF CONTINUITY. (NON-ARCHIMEDEAN GEOMETRY.)

CHAPTER III. 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 IV. THE THEORY OF PLANE AREAS.

§ 18. EQUAL AREA AND EQUAL CONTENT OF POLYGONS.

§ 19. PARALLELOGRAMS AND TRIANGLES HAVING EQUAL BASES AND EQUAL ALTITUDES.

§ 20. THE MEASURE OF AREA OF TRIANGLES AND POLYGONS.

§ 21. EQUALITY OF CONTENT AND THE MEASURE OF AREA.

CHAPTER V. 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 SYSTEM.

§ 29. CONSTRUCTION OF A GEOMETRY OF SPACE BY AID OF A DESARGUESIAN NUMBER SYSTEM.

§ 30. SIGNIFICANCE OF DESARGUES' S THEOREM.

CHAPTER VI. 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 VII. 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.

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