A survey of geometry

Cover......Page 1
Title page......Page 3
IX. ANALYTIC GEOMETRY I......Page 17
9.1 ANALYTIC GEOMETRY VERSUS SYNTHETIC GEOMETRY......Page 21
9.2 ABRIDGED NOTATION......Page 32
9.3 SOLVABILITY AND INSOLVABILITY WITH EUCLIDEAN TOOLS......Page 46
9.4 THE FUNDAMENTAL THEOREM OF LINEAR PROGRAMMING......Page 55
9.5 COORDINATE SYSTEMS; PLÜCKER'S DIMENSION THEORY......Page 64
9.6 REPRESENTATIONAL GEOMETRY......Page 75
10.1 HOMOGENEOUS COORDINATES AND INFINITE ELEMENTS......Page 85
10.2 LINE COORDINATES AND THE PRINCIPLE OF DUALITY......Page 92
10.3 IMAGINARY ELEMENTS AND THE CIRCULAR POINTS AT INFINITY......Page 96
10.4 PROJECTIVE COORDINATES IN A PLANE......Page 109
10.5 THE CONSISTENCY OF PLANE EUCLIDEAN GEOMETRY......Page 118
10.6 A CLASSIFICATION OF SOME ASSOCIATED GEOMETRIES......Page 123
10.7 MULTIPLE ALGEBRA AND VECTOR GEOMETRY......Page 129
XI. GEOMETRY AND THE THEORY OF GROUPS......Page 149
11.1 THE NOTION OF GROUP......Page 150
11.2 THE ERLANGER PROGRAMM......Page 159
11.3 ANALYTICAL ASPECT OF KLEINIAN GEOMETRIES......Page 165
11.4 PLANE AFFINE GEOMETRY......Page 173
11.5 CONCLUSION......Page 178
XII. GEOMETRY OF COMPLEX NUMBERS......Page 185
12.1 FUNDAMENTAL CONCEPTS AND OPERATIONS......Page 186
12.2 SOME ELEMENTARY GEOMETRY OF THE GAUSS PLANE......Page 192
12.3 ELEMENTARY TRANSFORMATIONS......Page 204
12.4 SOME GEOMETRY OF THE TRIANGLE......Page 211
12.5 CIRCULAR TRANSFORMATIONS......Page 216
12.6 CROSS RATIO......Page 227
XIII. LIMIT OPERATIONS IN GEOMETRY......Page 235
13.1 THE LIMIT CONCEPT IN ELEMENTARY GEOMETRY......Page 236
13.2 SOME GEOMETRICAL APPLICATIONS OF THE LIMIT CONCEPT......Page 243
13.3 SOME GEOMETRICAL APPLICATIONS OF THE CONTINUITY CONCEPT......Page 255
13.4 PATHOLOGICAL CURVES AND SURFACES......Page 266
XIV. DIFFERENTIAL GEOMETRY......Page 281
14.1 PLANE CURVES......Page 283
14.2 SPACE CURVES......Page 295
14.3 SURF ACES......Page 302
14.4 ANALYTICAL TREATMENT OF SPACE CURVES......Page 313
14.5 ANALYTICAL TREATMENT OF SURFACES......Page 330
14.6 CURVATURE OF SURFACES......Page 344
14.7 RIEMANNIAN GEOMETRY......Page 356
XV. COMBINATORIAL TOPOLOGY......Page 361
15.1 STANDARD SURFACES......Page 364
15.2 THE 2-MANIFOLDS......Page 375
15.3 CANONICAL FORMS AND THE CLASSIFICATION OF 2-MANIFOLDS......Page 383
15.4 ORIENTABLE AND NONORIENTABLE 2-MANIFOLDS, ONE-SIDED AND TWO-SIDED 2-MANIFOLDS......Page 391
15.5 EULER CHARACTERISTIC OF 2-MANIFOLDS......Page 398
15.6 RETICULATIONS ON 2-MANIFOLDS......Page 405
15.7 MAP-COLORING ON 2-MANIFOLDS......Page 410
15.8 VECTOR FIELDS......Page 421
16.1 n-DIMENSIONAL GEOMETRY......Page 431
16.2 METRIC SPACES......Page 438
16.3 HAUSDORFF AND TOPOLOGICAL SPACES......Page 448
16.4 CONTINUOUS MAPPINGS......Page 454
16.5 WHAT IS GEOMETRY?......Page 458
SUGGESTIONS FOR SOLUTIONS OF SOME OF THE PROBLEMS......Page 465
INDEX......Page 495