Geometry: Euclid and Beyond

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Springer Science & Business Media, 28 de set. 2005 - 528 pàgines
1 Ressenya
In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.
 

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Continguts

II
9
III
10
IV
20
V
29
VI
47
VII
53
VIII
67
IX
68
XXXIII
243
XXXV
244
XXXVI
252
XXXVII
261
XXXVIII
272
XXXIX
282
XL
297
XLI
298

X
75
XI
83
XII
92
XIII
98
XIV
106
XV
114
XVI
119
XVII
120
XVIII
130
XIX
137
XX
142
XXI
150
XXII
160
XXIII
167
XXIV
177
XXV
188
XXVI
197
XXVII
198
XXVIII
207
XXIX
214
XXX
223
XXXI
228
XXXII
233
XLII
306
XLIII
321
XLIV
328
XLV
336
XLVI
348
XLVII
357
XLVIII
375
XLIX
390
L
405
LI
417
LII
437
LIII
438
LIV
450
LV
460
LVI
471
LVII
483
LVIII
489
LIX
497
LX
505
LXI
507
LXII
509
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