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Guggenheimer H.W. — Plane geometry and its groups
Guggenheimer H.W. — Plane geometry and its groups

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Название: Plane geometry and its groups

Автор: Guggenheimer H.W.

Аннотация:

In this book the main theorems of plane euclidean geometry are proved on a university level. In the opinion of the writer, a reasonable training of high-school teachers must present them with a good working knowledge of geometry before starting to speak about foundations, i.e., about geometry.
The approach to geometry chosen in this book goes back to some papers by Ch. Wiener in the early 1890's. In particular also, I am indebted to the great book on plane geometry by Jacques Hadamard, and to modern works by G. Thomsen, H. Freudenthal, and F. Bachmann. Freudenthal has remarked that geometry today suffers from a low level of formalization as compared to other parts of mathematics. I am trying to remedy the situation by the use of a consistent formalism which is in keeping with the notations both of set theory and of traditional geometry. The deviations from the usual notations are kept to a minimum. Sometimes, however, they are unavoidable; for example, to untangle the different notions of angle that appear in elementary geometry, or, to distinguish between the point set of a segment and the vector which is its length.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1967

Количество страниц: 297

Добавлена в каталог: 26.07.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute, of hyperbolic geometry      225
Absolute, theorem      228
Addition points      106
Addition ratios      116
Addition vectors      42
Addition, of lengths      50
Addition, oriented angles      31
Addition, power series      260
Altitude      23
Anallagmatic      207
Angle bisector      9 33 263
Angle Bisector, external      16
Angle of analytic geometry      32
Angle of circles      149
Angle of pair of elliptic points      258
Angle of trigonometry      32 71
Angle Sum, hyperbolic      236
Angle, acute      20
Angle, alternating      37
Angle, center      89
Angle, elementary      9
Angle, obtuse      20
Angle, oriented      68 29 30
Angle, oriented, hyperbolic      231
Angle, right      20
Angle, straight      18
Angle, subtended at circumference      91
Angle, vertical      36
Antipodal point      251
Archimedes, axiom of      51
Archimedes, method of      197
Area function      107
Area Function of parallelogram      112
Area Function of rectangle      111
Area Function of triangle      112 124
Area Function, hyperbolic      237
Area Function, oriented      118
Artin, E.      266
Associative law      7 31 111
Asystatic      71 72
Automorphism of group      64
Automorphism of ordered field      199
Automorphism, of archimedean field      199
Axiom      I
Axis of symmetry      40
Axis, radical      148
Bachmann, F.      v 266
Bankoff, L.      277
Barrow, D.F.      186
Base      20
Bernstein, L.      277
BETWEEN      3 226
Blumenthal, L.      266
Boltyanskii, V.G.      121 266 267
Burckhardt, J.J.      266
Cantor, G.      285
Carlitz, L.      277
Center, of circle      85
Center, of group      64
Center, of homothety      152
Center, of hyperbolic circle      231
Center, of nine-point circle      165
Center, of rotation      29
Central      87
Centroid      48
Chord      89
circle      85
Circle of Appolonius      130
Circle of similitude      162
Circle, hyperbolic      231
Circle, M-      204
Circle, nine-point      165
Circumcenter      23
circumcircle      87
Circumradius      135
Closure, of plane      205
Collinear      4
Commutative law      31 43 61 111
Commutator subgroup      71
Commuting maps      12
Comparison theorems      19 199
Comparison, of angles      18
Comparison, of segments      17
Completeness      70
Concentric      89
concurrent      2
Concyclic      93
Congruence      10
Convex      4
Coordinates of hyperbolic line      247
Coordinates, homogeneous hyperbolic      247
Coordinates, in analytic geometry      262
Coset      65
Court, N.A.      266
Covering map      252
Coxeter, H.S.M.      266
Cross ratio      119 239 248
Curve of Constant Distance      233
De Comberousse, C.      267
Decagon, regular      137
Diameter      85
Disjoint      2
Distance      86
Distributive law      111
Duplication, of Cube      34
Edge, of polygon      44
Edge, of triangle      4
Element      2
END      229
Endpoint      3
Entity, elliptic      251
Entity, hyperbolic      225
Epimorphism      63
Equation of Line, normal      263
Equiangular Triangles      128
Equidistant      20
Equivalence, relation of      11
Erdoes, P.      186
Ergodic      184
Euclid      122
Euclidean algorithm      74
Euler line      50 165
Eves, H.      266
Excenter      88
Excircle      87
Exponential      63
Exponential of segment      248
Factor group      67
Faithful      71
Fejes Toth, L.      77 266
Field      116
Field of geometry      132
Field, non-Archimedean      136 262
Field, ordered      116
Field, Pythagorean      132
Field, uniquely ordered      136
Foot, of altitude      23
Forder, H.      266
Formal power series      260
Freudenthal, H.      v 266
Fundamental domain      73
Gauss plane      205
Geometry, analytic      116
Gerretsen, J.C.H.      266
Glide reflection      53
Glide Reflection, hyperbolic      233
Gnomon      107
Golden section      136
Gonseth, F.      266
Group      59
Group, abelian      61
Group, crystallographic      73
Group, crystallographic, hyperbolic      238
Group, cyclic      62
Group, cyclic, infinite      62
Group, GL(2, F)      248
Group, special linear      244
Hadamard, J.      v 266
Halfplane      3 262
Halfplane, hyperbolic      226
Harmonic range      129
Hart's Linkage      207
Heptagon      84
Hexagon, Pappus      99
Hexagon, Pascal      104
Hilbert Plane      260
Hilbert, D.      266
Hjelmslev's Midline      235
Hjelmslev, J.      10
Homogeneous space      66
Homomorphism      62
Homothety      152
Homothety of ratio one      153
Horocycle      233
Hypotenuse      24
Hypothesis      1
identity map      8
Incenter      24
incircle      87
Indeterminate      260
Inequality of L'Huilier      191
Inequality, arithmetic-geometric mean      187
Inequality, isoperimetric      194 195
Infinity, neighborhood of      205
Infinity, point at      205
Interior of convex set      86
Interior of polygon      72
Interior of triangle      5
Interior, of angle      18
Intersection      2
Inverse      10 59
inversion      200
Involution      8
Isogonal      28 95 143
Isometry      22
Isomorphism, of group      63
Isomorphism, of structure      224
Isoperimetric Polygons      189
Isotomic Map      40
Isotropy group      68
Jeger, M.      267
Kazarinoff, N.D.      267
Kernel, of map      71
Kernel, of set      7
Klein, F.      267
L'Huilier's Inequality      191
Lattice      73
Lattice group      73
Lattice, hexagonal      82
Lattice, parallelogram      79
Lattice, rectangular      80
Lattice, rhombic      80
Lattice, square      81
Leg, of angle      9
Leg, of isosceles triangle      20
Leg, of right triangle      24
Length of segment      50
Length, in real hyperbolic plane      240
Leuenberger, F.      275 277
Line      2 262
Line of support      85
Line, elliptic      252
Line, hyperbolic      225
Logarithmic function      63
Lune      257
M'Clelland, W.J.      267
MAP      7
Map, fractional linear      244
Matrix      244
Matrix, product      244
Median      23
Method, axiomatic      1
Midline      9 264
Midline, Hjelmslev's      235
midpoint      9
Midpoint of edge      44 45
Midpoint of hyperbolic lines      234
Mitrinovic, D.S.      277
Model, of a geometry      224
Moebius Circle      204
Moise, E.E.      267
Monomorphism      63
Mordell, L.J.      186
Moser, W.O.      266
Motion, euclidean      10 60
Motion, horocyclic      233
Motion, proper      60
Multiplication, of lengths      111
Multiplication, of oriented segments      114
Multiplication, of points      106
Multiplication, of power series      260
Multiplication, of ratios      116
Nairizi, An      121
Nowacki, W.      77
Null set      2
Numbers, mod n      61
Numbers, real      61
Order, of cyclic group      62
Origin, of coordinates      116
Ornament      72
Ornament, ribbon      84
orthocenter      47
Pappus, problem of      197
Pappus, theorem      138
Pappus, theorem, affine      100
parallel      2
Parallel, angle      240 248
Parallel, polygon, inner      191
Parallelogram      38
Parallelogram law      43
Parallelogram, complementary      107
Pascal line      104 145
Path, regular      182
Peaucellier's cell      207
Pedoe, D.      277
Pencil, maximal      26
Pencil, of circles      148
Pencil, of lines      26
Pencil, of M-circles      209
Pentagon, regular      136
Perpendicular      9
Perpendicular bisector      9 263
Perspective      101
Plane, elliptic      252
Plane, Gauss      205
Plane, Hilbert      260
Plane, hyperbolic      225
Point      2 66
Point at infinity      26 205
Point, antipodal      251
Point, diagonal      100
Point, elliptic      251
Point, Feuerbach      169
Point, ideal      26
Point, invariable      163
Point, negative Brocard      170
Point, positive Brocard      170
Point, symmedian      169
polar      207
Polar of elliptic point      255
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