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Foundations of Geometry: Euclidean, Bolyai-Lobachevskian, and Projective Geometry: Euclidean, Bolyai-Lobachevskian, and Projective Geometry [Minkštas viršelis]

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  • Formatas: Paperback / softback, 448 pages, aukštis x plotis x storis: 230x152x24 mm, weight: 650 g
  • Išleidimo metai: 28-Dec-2018
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486828093
  • ISBN-13: 9780486828091
Kitos knygos pagal šią temą:
Foundations of Geometry: Euclidean, Bolyai-Lobachevskian, and Projective Geometry: Euclidean, Bolyai-Lobachevskian, and Projective GeometryDidesnis paveikslėlis
  • Formatas: Paperback / softback, 448 pages, aukštis x plotis x storis: 230x152x24 mm, weight: 650 g
  • Išleidimo metai: 28-Dec-2018
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486828093
  • ISBN-13: 9780486828091
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780486828091.html
Raktažodžiai:
Part One of this comprehensive treatment develops Euclidean and Bolyai-Lobachevskian geometry on the basis of a Hilbert system, and Part Two explores projective geometry in much the same way. 1960 edition.


In Part One of this comprehensive and frequently cited treatment, the authors develop Euclidean and Bolyai-Lobachevskian geometry on the basis of an axiom system due, in principle, to the work of David Hilbert. Part Two develops projective geometry in much the same way. An Introduction provides background on topological space, analytic geometry, and other relevant topics, and rigorous proofs appear throughout the text.
Topics covered by Part One include axioms of incidence and order, axioms of congruence, the axiom of continuity, models of absolute geometry, and Euclidean geometry, culminating in the treatment of Bolyai-Lobachevskian geometry. Part Two examines axioms of incidents and order and the axiom of continuity, concluding with an exploration of models of projective geometry. 
Preface To The Polish Edition xi
Preface To The English Edition xiii
Introduction
1 Geometry before Euclid
1(1)
2 Elements of Euclid
1(1)
3 Elementary Geometry after Euclid. Euclid's Critics and Commentators
2(2)
4 Bolyai-Lobachevskian Geometry
4(1)
5 Consistency of Geometry
4(2)
6 Riemann Spaces
6(1)
7 Axiomatic Theory
6(1)
8 Sets and Relations
7(4)
9 Topological Space
11(4)
10 Analytic Geometry
15(4)
Part One: Euclidean And Bolyai-Lobachevskian Geometry
Introduction To Part One: Primitive Notions and Axioms
19(2)
Chapter I Axioms Of Incidence And Order
1 Axioms of Incidence
21(1)
2 Three Non-Collinear Points and Four Non-Coplanar Points
22(1)
3 Lines and Planes
22(1)
4 Fundamental Existence Theorems
23(2)
5 Intersections of Lines and Planes
25(1)
6 Linear Axioms of Order
26(1)
7 Segments. Open Segments
27(1)
8 Open Segments on a Line
28(2)
9 Division of an Open Segment
30(1)
10 Common Part of Two Open Segments
31(1)
11 Topology on the Line
32(1)
12 Half-Lines
33(3)
13 Half-Lines on a Line
36(1)
14 Half-Lines on a Given Half-Line
37(1)
15 Orientations of a Line. Axes
37(2)
16 Order of Points on an Axis
39(2)
17 Betweenness Relation for a Line and Two Points
41(1)
18 Plane Axiom of Order
42(1)
19 Half-Planes
43(4)
20 Pencils and Half-Pencils of Half-Lines
47(3)
21 Betweenness Relation for Half-Lines
50(5)
22 Ordering of a Half-Pencil
55(1)
23 Angles
55(2)
24 Characterization of Lines end Planes in Terms of the Betweenness Relation
57(1)
25 Triangles. Open Triangles
58(3)
28 The Open Triangle and the Line
61(3)
27 Topology on the Plane
64(4)
28 Polygonal Lines
68(2)
29 Quadrangles
70(1)
30 The Intersection Point of the Diagonals of a Quadrangle
71(5)
31 Polygons and Their Triangulation
76(4)
Chapter II Axioms Of Congruence
1 Axioms of Congruence. Congruence of Segments. Congruence of Figures
80(2)
2 Relations Between Segments on Two Lines
82(1)
3 Relations Between Segments on Two Planes
83(1)
4 Congruence of Angles
84(2)
5 Adjacent Angles. Vertical Angles
86(2)
6 Relations Between Angles of Two Half-Pencils
88(2)
7 Relations Between Sides and Angles of Two Triangles
90(1)
8 Relations Between Sides and Angles of a Triangle
91(1)
9 The Relations Less-Than and Greater-Than for Segments
92(2)
10 The Relations Less-Than and Greater-Than for Angles
94(1)
11 Midpoint of a Segment
95(2)
12 Bisector of an Angle
97(1)
13 External Angles of a Triangle
97(1)
14 Two Non-Intersecting Lines on a Plane
98(2)
15 Relations Between Sides and Angles of a Triangle (Conclusion)
100(1)
16 Relations Between Sides and Angles of Two Triangles (Continued)
101(2)
17 Free Segments. The Relations Less-Than and Greater-Than for Free Segments. Addition of Free Segments
103(2)
18 Subtraction of Free Segments. Multiplication of Free Segments by Dyadic Numbers
105(2)
19 Triangle Inequality
107(2)
20 Free Angles. Calculus of Free Angles
109(3)
21 Addition of Free Angles in a Pencil
112(1)
22 The Sum of Two Angles of a Triangle
113(1)
23 Right, Acute, and Obtuse Angles
114(2)
24 Right, Acute, and Obtuse Free Angles
116(1)
25 Right, Acute, and Obtuse Triangles
117(2)
26 Perpendicular Lines
119(1)
27 Perpendicular Projection upon a Line
120(1)
28 Perpendicular Bisector of a Segment
121(1)
29 The Saccheri Quadrangle
121(2)
30 Rectangels
123(1)
31 Line Perpendicular to a Plane
124(5)
32 Perpendicular Planes
129(1)
33 Perpendicular Projection upon a Plane
130(1)
34 Congruence of Two Lines and Two Planes. Perfect Homogeneity of the Line and of the Plane
131(7)
35 Parallel Half-Lines
138(11)
36 Parallel Axes
149(1)
37 Parallel Lines
150(1)
Chapter III Axiom Of Continuity
1 Axiom of Continuity
151(3)
2 The Archimedean Postulate
154(3)
3 The Saceheri Quadrangle (Conclusion)
157(1)
4 The Saccheri-Legendre Theorem
158(2)
5 Parallel Half-Lines (Continued)
160(1)
6 Parallel Axes (Continued)
161(1)
7 Angle of Parallelism
161(4)
8 Parallel Lines (Continued)
165(2)
9 Measure of Segments
167(5)
10 Measure of Angles
172(2)
11 The Saccheri-Legendre Theorem Formulated in Terms of Measure
174(2)
12 Distance Between Two Points. Space S as a Metric Space
176(1)
13 Distance of a Point from a Figure
176(1)
14 A Characterization of Betweenness and Equidistance Relations in Terms of Distance
177(1)
15 Similitudes
177(2)
16 Topology in Space Induced by a Metric
179(2)
17 Distance as a Continuous Function of Two Points
181(3)
18 Coordinates on a Line. Metric Type of Lines
184(1)
19 Absolute Coordinates on a Plane. Topological Type of Planes
185(4)
20 Absolute Coordinates in Space. Topological Type of Space
189(3)
21 Rectangular Coordinates
192(2)
Chapter IV Models Of Absolute Geometry
1 Problems of Consistency, Independence, and Categoricity of an Axiom System of Geometry. Interpretation. Model
194(3)
2 The Cartesian Space Cn
197(14)
3 The Cartesian Model. Consistency of Absolute Geometry
211(7)
4 Independence of the Axiom of Continuity
218(7)
5 The Cartesian Circle
225(7)
6 Projective Space Pn
232(13)
7 The Klein-Beltrami Model
245(14)
8 Formula for Distance in Klein Space K2
259(3)
9 Non-Categoricity of Absolute Geometry. Euclidean Geometry. Bolyai Lobachevskian Geometry
262(2)
Chapter V Euclidean Geometry
1 The Axiom of Euclid
264(1)
2 The Sum of the Angles of a Triangle and of a Quadrangle
264(1)
3 Parallel Lines (Conclusion)
264(1)
4 Parallel Half-Lines (Conclusion)
265(2)
5 Two Angles with Respectively Parallel Sides
267(1)
6 Parallel Projection upon a Line
268(1)
7 Parallelograms
268(1)
8 The Theorem of Thales
269(3)
9 Similar Triangles
272(1)
10 Pythagorean Theorem
273(1)
11 Metric Type of Planes
273(1)
12 Metric Type of Space
274(2)
13 Categoricity of Euclidean Geometry
276(2)
Chapter VI Bolyai-Lobachevskian Geometry
1 The Axiom of Bolyai-Lobachevski
278(1)
2 The Sum of the Angles of a Triangle and of a Quadrangle
278(1)
3 Relations Between Sides and Angles of Two Triangles (Conclusion)
279(1)
4 Similitudes
280(1)
5 Defect of a Triangle
280(3)
6 Defect of a Polygon
283(1)
7 Defect of a Plane Figure
284(1)
8 Parallel Lines and Hyperparallel Lines
284(2)
9 Pairs of Parallel Lines
286(4)
10 Pairs of Hyperparallel Lines
290(3)
11 Perpendicular Projection of a Line Upon a Line
293(5)
12 The Line of Enclosure
298(2)
13 Natural Basic Segment. Natural Measure of Segments
300(1)
14 Three Parallel Lines
301(1)
15 Perpendicular Bisectors of the Three Sides of a Triangle
302(2)
16 The Lobachevskian Function II
304(1)
17 The Infinite Right Triangle
305(3)
18 The Horocycle
308(8)
19 Tangents and Secants of the Horocycle
316(2)
20 Arc of the Horocycle
318(1)
21 Length of Arc of the Horocycle
319(4)
22 Translations on a Plane. Length of Arc on a Translated Horocycle
323(4)
23 Length of Arc of the Horocycle (Conclusion)
327(2)
24 Sectors of the Horocycle
329(2)
25 Formula of the Lobachevskian Function
331(3)
26 The Functions sin II and cos II
334(1)
27 Right Triangles
335(3)
28 Quadrangle with Three Right Angles
338(1)
29 Rectangular Coordinates on a Plane
339(2)
30 Beltrami Coordinates on a Plane. Formula for Distance
341(3)
31 Categoricity of Bolyai-Lobachevskian Geometry
344(5)
Part Two: Projective Geometry
Introduction To Part Two: Primitive Notions and Axioms
349(1)
Chapter VII Axioms Of Incidence And Order
1 Axioms of Incidence
350(1)
2 Fundamental Existence Theorems
351(2)
3 Intersections of Lines and Planes
353(1)
4 Central Projection upon a Line. Perspective and Projective Transformations
354(1)
5 Central Projection upon a Plane
355(1)
6 Triangles. Perspective Center and Perspective Axis of Two Triangles The Theorem of Desargues
356(9)
7 Axioms of Order
365(2)
8 Segments. Open Segments
367(1)
9 Properties of Open Segments
368(2)
10 The Triple of Open Sides of a Triangle
370(1)
11 Model for the Euclidean Axioms of Incidence and Order in Projective Geometry. Proper Points, Lines, and Planes
370(3)
12 Ordinary Triangles
373(2)
13 Topology on the Line
375(1)
14 Topology on the Plane
376(1)
15 Quadrangles
377(2)
16 Harmonic Quadruples
379(1)
17 Permutations of Harmonic Quadruples
380(2)
18 The Fourth Harmonic Point
382(4)
19 Perspective Transformations of Harmonic Quadruples
386(1)
20 Continuity of the Central Projectivity
387(1)
21 Midpoint of a Segment
388(1)
22 Natural Net
388(3)
23 Integral Net
391(1)
24 Dyadic Net
392(3)
Chapter VIII Axiom Of Continuity
1 Axiom of Continuity
395(1)
2 Dyadic Net (Conclusion)
396(2)
3 Real Net
398(3)
4 Cartesian Coordinates on a Line
401(1)
5 Cartesian Coordinates on a Plane
401(4)
6 Equation of the Set of Proper Points of a Proper Line on a Plane
405(3)
7 Cartesian Coordinates in Space
408(1)
8 Equation of the Set of Proper Points of a Proper Line in Space
408(1)
9 Equation of the Set of Proper Points of a Proper Plane in Space
409(2)
10 Projective Coordinates in Space
411(1)
11 Equations of the Plane and of the Line in Space
412(2)
12 The Relation of Division and the Cross Ratio
414(3)
Chapter IX Models Of Projective Geometry
1 Problems of Consistency and Categoricity
417(1)
2 Model (P). Consistency of Projective Geometry
418(2)
3 Categoricity of Space Projective Geometry
420(1)
4 The Ellipse. Some Theorems Concerning the Circle and the Ellipse
421(4)
5 The Limit of a Sequence of Circles
425(3)
6 Problem of the Categoricity of Plane Projective Geometry. The Hilbert Model
428(9)
Index Of Geometrical Symbols 437(3)
Index 440