# Geometry: Euclid and Beyond

Springer Science & Business Media, Sep 28, 2005 - Mathematics - 528 pages
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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### Contents

 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 Copyright