Ideas of space: Euclidean, non-Euclidean, and relativistic
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
1989
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 242 S. graph. Darst. |
| ISBN: | 0198539355 0198539347 |
Staff View
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| 245 | 1 | 0 | |a Ideas of space |b Euclidean, non-Euclidean, and relativistic |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Oxford |b Clarendon Press |c 1989 | |
| 300 | |a XI, 242 S. |b graph. Darst. | ||
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| adam_text | Contents
Part 1
1 Early geometry 1
The contrast between Greek and Babylonian mathematics raised. The
limitations of rhetorical algebra. The earliest Greek mathematics, a search
for deductive validity. Pythagoras theorem known to the Babylonians,
c. 1700 bc; a dissection proof based on the theory of figured numbers.
Incommensurable magnitudes shown to exist; a conjectural account of
their discovery. Reasons for the retention of geometry discussed. The
discovery of the irrationality of y/2 led to further research into
incommensurability, but not a foundational crisis. Similarity and parallel¬
ism. Eudoxus on the nature of ratio. Exercises. Pythagorean triples, Greek
and Babylonian methods. Plimpton 322
2 Euclidean geometry and the parallel postulate 26
Geometry as the study of figures in space, and assumptions about space.
Euclid s Elements. The parallel postulate; is such an assumption necessary?
Consequences of the postulate; existence and uniqueness of parallels;
equidistant lines. Various arguments to prove the postulate, including
those of Proclus. Assumptions equivalent to the parallel postulate.
Appendix on solid geometry and trigonometry. A geometry on the sphere.
Plane and spherical trigonometry. Exercises
3 Investigations by Islamic mathematicians 41
Early Arabic mathematics, al Khwarizimi. Investigators of the parallel
postulate: al Gauhari; Thabit ibn Qurra and the line generated by motion
along a straight line; ibn al Haytham s attempt using motion; Omar
Khayyam s critique of the use of motion in geometry, and his attempt using
a postulate of Aristotle s; Nasir Eddin al Tusi s approach. The influential
approach attributed to al Tusi by Western writers. Exercises
Parti
4 Saccheri and his Western predecessors 57
Revival of interest in the West in the 16th and 17th centuries. The work of
Saccheri. Three hypotheses. HOA refuted. HAA discussed. Exercises
5 J. H. Lambert s work 70
Spherical geometry. The work of J. H. Lambert. Absolute nature of length.
Angle sum and area. The imaginary sphere. Exercises
x Contents
6 Legendre s work 78
French lack of interest; except for Fourier and Legendre. Legendre s
refutations of non Euclidean geometries. Exercises
7 Gauss s contribution 83
Kant. Gauss s work. Directed lines. A new definition of parallel.
Corresponding points. The horocycle. Exercises
8 Trigonometry 92
Trigonometric and hyperbolic functions. Exercises
9 The first new geometries 97
Schweikart s Astral Geometry. Taurinus s logarithmic spherical geo¬
metry. Approximate agreement between the new geometries and the old.
Appendix. Exercises
10 The discoveries of Lobachevskii and Bolyai 106
The Bolyai s struggle. Absolute geometry. The work of Lobachevskii
summarized and described. The prism theorem; the horocycle and
horosphere; the projection map. Geometry on the horosophere is
Euclidean; the fundamental formulae of hyperbolic geometry. Bolyai s
work; squaring the circle. Summary. Priorities. Exercises. Appendix on
spherical trigonometry, including Bolyai s proof of its absolute nature
11 Curves and surfaces 129
Curves. Surfaces; co ordinates; curvature. Intrinsic and extrinsic view¬
points. Geodesies. Minding s surface. Appendix on degeneracies
12 Riemann on the foundations of geometry 141
Riemann s hypotheses. Co ordinates on surfaces. Intrinsic geometry and
curvature; metrical ideas at the basis of geometry
13 BeUrami s ideas 147
Beltrami s map. Relative consistency of mathematics; foundational
questions. Bolyai Lobachevskii formulae
14 New models and old arguments 155
Klein s model of elliptic geometry. Poincare s conformal model of
hyperbolic geometry. Revisiting the work of Wallis, Saccheri, and
Legendre. Balzac. Exercises
15 Resume 168
Summary of Part 2 and other views
Contents xi
Part 3
16 Non Euclidean mechanics 175
Dostoevsky. Non Euclidean mechanics
17 The question of absolute space 177
Newton; Newtonian space. Relative motion. Magnetism and electricity.
Ether drift?; absolute space? Einstein s idea. Kennedy Thorndike experi¬
ment. The nature of space
18 Space, time, and space time 190
Space time. Clocks and surveying. The invariance of distance. Change of
axes. Invariance of the interval. Summary. Appendix on co ordinate
transformations. Exercises
19 Paradoxes of special relativity 204
The paradoxes of special relativity posed and solved
20 Gravitation and non Euclidean geometry 210
Gravity, its relative nature. The conventional element in measurement.
The heated plate and the cooled plate universes, their connections with
non Euclidean geometries. The rubber sheet model of gravity. Exercises
21 Speculations 218
Gravitation in four dimensional space time, curvature, black holes.
Speculations. Appendix, W. K. Clifford
22 Some last thoughts 226
Meanings. Mathematical appendix on the connection between non
Euclidean geometry and special relativity, and on transformation groups
List of mathematicians and physicists 233
Bibliography 235
Index 240
|
| adam_txt |
Contents
Part 1
1 Early geometry 1
The contrast between Greek and Babylonian mathematics raised. The
limitations of rhetorical algebra. The earliest Greek mathematics, a search
for deductive validity. Pythagoras' theorem known to the Babylonians,
c. 1700 bc; a dissection proof based on the theory of figured numbers.
Incommensurable magnitudes shown to exist; a conjectural account of
their discovery. Reasons for the retention of geometry discussed. The
discovery of the irrationality of y/2 led to further research into
incommensurability, but not a foundational crisis. Similarity and parallel¬
ism. Eudoxus on the nature of ratio. Exercises. Pythagorean triples, Greek
and Babylonian methods. Plimpton 322
2 Euclidean geometry and the parallel postulate 26
Geometry as the study of figures in space, and assumptions about space.
Euclid's Elements. The parallel postulate; is such an assumption necessary?
Consequences of the postulate; existence and uniqueness of parallels;
equidistant lines. Various arguments to 'prove' the postulate, including
those of Proclus. Assumptions equivalent to the parallel postulate.
Appendix on solid geometry and trigonometry. A geometry on the sphere.
Plane and spherical trigonometry. Exercises
3 Investigations by Islamic mathematicians 41
Early Arabic mathematics, al Khwarizimi. Investigators of the parallel
postulate: al Gauhari; Thabit ibn Qurra and the line generated by motion
along a straight line; ibn al Haytham's attempt using motion; Omar
Khayyam's critique of the use of motion in geometry, and his attempt using
a postulate of Aristotle's; Nasir Eddin al Tusi's approach. The influential
approach attributed to al Tusi by Western writers. Exercises
Parti
4 Saccheri and his Western predecessors 57
Revival of interest in the West in the 16th and 17th centuries. The work of
Saccheri. Three hypotheses. HOA refuted. HAA discussed. Exercises
5 J. H. Lambert's work 70
Spherical geometry. The work of J. H. Lambert. Absolute nature of length.
Angle sum and area. The imaginary sphere. Exercises
x Contents
6 Legendre's work 78
French lack of interest; except for Fourier and Legendre. Legendre's
'refutations' of non Euclidean geometries. Exercises
7 Gauss's contribution 83
Kant. Gauss's work. Directed lines. A new definition of parallel.
Corresponding points. The horocycle. Exercises
8 Trigonometry 92
Trigonometric and hyperbolic functions. Exercises
9 The first new geometries 97
Schweikart's Astral Geometry. Taurinus's logarithmic spherical geo¬
metry. Approximate agreement between the new geometries and the old.
Appendix. Exercises
10 The discoveries of Lobachevskii and Bolyai 106
The Bolyai's struggle. Absolute geometry. The work of Lobachevskii
summarized and described. The prism theorem; the horocycle and
horosphere; the projection map. Geometry on the horosophere is
Euclidean; the fundamental formulae of hyperbolic geometry. Bolyai's
work; squaring the circle. Summary. Priorities. Exercises. Appendix on
spherical trigonometry, including Bolyai's proof of its absolute nature
11 Curves and surfaces 129
Curves. Surfaces; co ordinates; curvature. Intrinsic and extrinsic view¬
points. Geodesies. Minding's surface. Appendix on degeneracies
12 Riemann on the foundations of geometry 141
Riemann's hypotheses. Co ordinates on surfaces. Intrinsic geometry and
curvature; metrical ideas at the basis of geometry
13 BeUrami's ideas 147
Beltrami's map. Relative consistency of mathematics; foundational
questions. Bolyai Lobachevskii formulae
14 New models and old arguments 155
Klein's model of elliptic geometry. Poincare's conformal model of
hyperbolic geometry. Revisiting the work of Wallis, Saccheri, and
Legendre. Balzac. Exercises
15 Resume 168
Summary of Part 2 and other views
Contents xi
Part 3
16 Non Euclidean mechanics 175
Dostoevsky. Non Euclidean mechanics
17 The question of absolute space 177
Newton; Newtonian space. Relative motion. Magnetism and electricity.
Ether drift?; absolute space? Einstein's idea. Kennedy Thorndike experi¬
ment. The nature of space
18 Space, time, and space time 190
Space time. Clocks and surveying. The invariance of distance. Change of
axes. Invariance of the interval. Summary. Appendix on co ordinate
transformations. Exercises
19 Paradoxes of special relativity 204
The 'paradoxes' of special relativity posed and solved
20 Gravitation and non Euclidean geometry 210
Gravity, its relative nature. The conventional element in measurement.
The heated plate and the cooled plate universes, their connections with
non Euclidean geometries. The rubber sheet model of gravity. Exercises
21 Speculations 218
Gravitation in four dimensional space time, curvature, black holes.
Speculations. Appendix, W. K. Clifford
22 Some last thoughts 226
Meanings. Mathematical appendix on the connection between non
Euclidean geometry and special relativity, and on transformation groups
List of mathematicians and physicists 233
Bibliography 235
Index 240 |
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| index_date | 2024-07-02T16:04:20Z |
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| spelling | Gray, Jeremy Verfasser aut Ideas of space Euclidean, non-Euclidean, and relativistic 2. ed. Oxford Clarendon Press 1989 XI, 242 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Espace et temps - Modèles mathématiques - Histoire Espace-temps - Modèles mathématiques - Histoire ram Géométrie - Histoire Géométrie - Histoire ram Meetkunde gtt Wiskundige modellen gtt Geschichte Mathematisches Modell Geometry History Space and time Mathematical models History Raum Mathematik (DE-588)4124030-3 gnd rswk-swf Nichteuklidische Geometrie (DE-588)4042073-5 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Raumvorstellung (DE-588)4121556-4 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Euklidische Geometrie (DE-588)4137555-5 gnd rswk-swf Nichteuklidische Geometrie (DE-588)4042073-5 s DE-604 Raum Mathematik (DE-588)4124030-3 s Euklidische Geometrie (DE-588)4137555-5 s Geometrie (DE-588)4020236-7 s Raumvorstellung (DE-588)4121556-4 s Geschichte (DE-588)4020517-4 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015110167&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gray, Jeremy Ideas of space Euclidean, non-Euclidean, and relativistic Espace et temps - Modèles mathématiques - Histoire Espace-temps - Modèles mathématiques - Histoire ram Géométrie - Histoire Géométrie - Histoire ram Meetkunde gtt Wiskundige modellen gtt Geschichte Mathematisches Modell Geometry History Space and time Mathematical models History Raum Mathematik (DE-588)4124030-3 gnd Nichteuklidische Geometrie (DE-588)4042073-5 gnd Geometrie (DE-588)4020236-7 gnd Raumvorstellung (DE-588)4121556-4 gnd Geschichte (DE-588)4020517-4 gnd Euklidische Geometrie (DE-588)4137555-5 gnd |
| subject_GND | (DE-588)4124030-3 (DE-588)4042073-5 (DE-588)4020236-7 (DE-588)4121556-4 (DE-588)4020517-4 (DE-588)4137555-5 |
| title | Ideas of space Euclidean, non-Euclidean, and relativistic |
| title_auth | Ideas of space Euclidean, non-Euclidean, and relativistic |
| title_exact_search | Ideas of space Euclidean, non-Euclidean, and relativistic |
| title_exact_search_txtP | Ideas of space Euclidean, non-Euclidean, and relativistic |
| title_full | Ideas of space Euclidean, non-Euclidean, and relativistic |
| title_fullStr | Ideas of space Euclidean, non-Euclidean, and relativistic |
| title_full_unstemmed | Ideas of space Euclidean, non-Euclidean, and relativistic |
| title_short | Ideas of space |
| title_sort | ideas of space euclidean non euclidean and relativistic |
| title_sub | Euclidean, non-Euclidean, and relativistic |
| topic | Espace et temps - Modèles mathématiques - Histoire Espace-temps - Modèles mathématiques - Histoire ram Géométrie - Histoire Géométrie - Histoire ram Meetkunde gtt Wiskundige modellen gtt Geschichte Mathematisches Modell Geometry History Space and time Mathematical models History Raum Mathematik (DE-588)4124030-3 gnd Nichteuklidische Geometrie (DE-588)4042073-5 gnd Geometrie (DE-588)4020236-7 gnd Raumvorstellung (DE-588)4121556-4 gnd Geschichte (DE-588)4020517-4 gnd Euklidische Geometrie (DE-588)4137555-5 gnd |
| topic_facet | Espace et temps - Modèles mathématiques - Histoire Espace-temps - Modèles mathématiques - Histoire Géométrie - Histoire Meetkunde Wiskundige modellen Geschichte Mathematisches Modell Geometry History Space and time Mathematical models History Raum Mathematik Nichteuklidische Geometrie Geometrie Raumvorstellung Euklidische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015110167&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT grayjeremy ideasofspaceeuclideannoneuclideanandrelativistic |