The King of Infinite Space (17 page)

Associative laws,
105

Associative operation,
142

Assumptions,
11
,
12
,
27
,
45
,
46
,
55
,
83
,
119

of existence of points/lines/planes,
49
,
107

hidden,
30–31

that the parallel postulate is false,
120
,
121

Atiyah, Michael,
91

Atoms,
41–42
,
43
,
44

Axiomatic systems,
11
,
12
,
14
,
149

and arguments,
17

new,
107

as way of life,
9
,
106
,
148
,
152
,
156

Axioms,
45–56
,
80
,
90
,
152

Archimedean axiom,
108–109

Cantor-Dedekind axiom,
100

completeness axiom,
108
,
109

of connection, order, congruence and continuity,
107

consistency/inconsistency of,
106–107
,
131

for fields,
104–105
,
112

fifth axiom,
53–56
,
118
(
see also
Axioms: Playfair's axiom
;
Parallel postulate
)

first axiom,
113–114

first three axioms,
46
,
48–49
,
51
,
61
,
66
,
86

fourth axiom,
50–51
,
73

fourth proposition as axiom,
27

Hilbert's axioms,
109
,
111

interpreted in arithmetic,
113–114

made theorems,
46

of neutral geometry,
131

Playfair's axiom,
54
,
55
,
91
,
137
,
138

relationship between axioms and theorems,
12
,
14
,
19
,
149

as self-evident,
46

See also
Axiomatic systems

Babylonians,
8
,
69

Bacon, Francis,
77

Beltrami, Eugenio,
121
,
132–133

Bolyai, János,
118
,
122–123
,
126
,
127–128

Bolyai, Farkas,
127–128

Boole, George,
23

Boundaries,
160

Breadth,
33
,
35
,
36
,
159

Bridge of Asses,
64
,
65(fig.)
.
See also
Propositions: fifth proposition

Calculus,
39
,
94

differential calculus,
41
,
59

of segments,
110

Cantor, Georg,
93

Cantor-Dedekind axiom,
100
,
101
,
109

Cardioids,
99
,
99(fig.)

Cathedrals,
64

Causality,
13

Cézanne, Paul,
152

Change,
43
,
44
,
52

Chesterton, G. K.,
11

China,
9

Cicero,
1

Circles,
7
,
13
,
25
,
46
,
79

center/circumference of,
98
,
130
,
135
,
136
,
160

diameter of,
160

and geodesics,
125

and proposition one,
61–62

radii of,
49
,
62
,
98
,
130

semicircles,
160

See also
Poincaré, Henri: Poincaré disk

Clark, Kenneth,
78
,
152

Clay tablets,
8

Coincidence,
21
,
23
,
25–26
,
27
,
39
,
41
,
67

and concrete vs. abstract models of geometry,
28–29

Common beliefs/notions,
19–32
,
90

fifth,
29–30

first,
24

fourth,
23

second/third,
24
,
62
,
74

Common sense,
36
,
64
,
91
,
118
,
124
,
130
,
139

Commutative laws,
105

Compass,
63
.
See also
Straight-edge and compass

Complexity,
55
,
107

Computers,
150

Congruence,
26
,
39
,
67
,
73
,
74
,
75
,
107
,
130

Consistency/inconsistency,
106–107
,
131

Contradictions,
17
,
83
,
87
,
89
,
100
,
120
,
121
,
131
.
See also
Reductio ad absurdum

Contrapositives,
83
,
84(n)
,
86
,
86(fig.)

Converse relationship,
69
,
81(n)
,
82
,
83

Coordinate Method, The
(Gelfand, Glagoleva, and Kirillov),
99–100

Coordinate systems,
97
,
97(fig.)
,
115

Critique of Pure Reason, The
(Kant),
117

Cultures,
3
,
4
,
9

Curvature,
38
,
99
,
125
,
139

extrinsic,
40
,
41

negative,
133

and straight lines,
39

Das Kontinuum
(Weyl),
44

Davies, Brian,
150
,
151

De Architectura
(Vitruvius Pollio),
1–2

Dedekind, Richard,
102
.
See also
Cantor-Dedekind axiom

Deduction,
45
,
149

Definitions,
20
,
33–44
,
51
,
90
,
159–161

eighth and ninth,
51–53

fifteenth, sixteenth, and seventeenth,
62

fifth,
35

first seven and twenty-third,
33–34

fourth,
38

of hyperbolic lines/distance,
134–135
,
136
,
139

nineteenth,
60
,
84
,
85–86

ninth through twenty-second,
34

and real ordered fields,
113

of rectilinear figures,
60

seventh,
38

of shape,
49

tenth,
73

third,
43

twentieth,
60

twenty-third,
37–38
,
43
,
84

Degrees of freedom,
37

Democritus,
41
,
42
,
44

De Morgan, August,
84(n)

Descartes, René,
45
,
96

Dieudonné, Jean,
115

Dimensions,
35
,
36
,
37
,
40
,
70
,
125
,
141
,
144

Distance,
23
,
37
,
39
,
41
,
56
,
69–70
,
87
,
88
,
125
,
132
,
144

hyperbolic distance,
135–137
,
139

Distributive laws,
105

Division,
93
,
95
,
103
,
104
,
110
,
112

Egyptians,
11

Einstein, Albert,
118

Elementary Geometry from an Advanced Standpoint
(Moise),
94

Elements
(Euclid),
9
,
44
,
80
,
90
,
91
,
123
,
153

Book I,
6
,
119

Book II,
6

Book V,
7
,
93–94
,
110

Books V through IX,
7

Book VII,
93
,
110

Book X,
93–94
,
100
,
110

books in,
6–7

first four books,
7

as having limited symbolic reach,
71

as illustrated,
59
,
64–65
,
79–80
,
87
,
90

and mountain-climbing pastoral,
57–58

as textbook,
5–6
,
7
,
155–156

Eliot, George,
45

Ellipses,
13
,
98–99

Empson, William,
57–58

Encyclopedia Britannica
,
28

Equality,
21
,
22–25
,
26
,
36
,
62
,
63
,
148

definition of,
25

“less than or equal to,”
105

of right angles,
50–51

of squares,
75

transitivity of,
24

See also
Angles: as equal

Equator,
125

Erlangen program,
140
,
142

Ethics,
123

Euclid,
21–22
,
43
,
89
,
140
,
145
,
152–153

and Aristotle,
15
,
17

birth/death of,
5

double insight of,
12

Euclidean ideal,
150

Euclidean style,
148–149

Euclidean tradition,
155–156

and fifth axiom (parallel postulate),
54–55
,
118–119
,
139–140

as a mathematician,
6

modern versions of,
8

predecessors of,
6

as a teacher,
5–6
,
26–27
,
79

translations of,
8

and unity beneath diversity of experience,
11

Euclides ab omni naevo vindicatus
(Saccheri),
121

Eudoxus,
6
,
94
,
108

Explicit (word),
149

Fields,
103–106
,
112
,
113
,
118
,
142

Flatness,
38–39
,
40–41

Flaubert, Gustave,
1

Forms (Platonic),
13
,
60
,
145

Four-color theorem,
151

Fractions.
See under
Numbers

Friedman, Harvey,
46

Galois, Évariste,
142

Gauss, Carl Friedrich,
41
,
48
,
92
,
93
,
118
,
122
,
126
,
127

Gelfand, I. M.,
99

Geodesics,
125
,
126
,
137

Geometry,
5
,
6
,
12
,
80
,
83
,
112

analytic geometry,
96–97
,
98–100
,
108
,
109
,
110
,
115

classification of geometries,
140–142

concrete vs. abstract models of,
13–14
,
28–29

differential geometry,
41

elliptical geometry,
141

Euclidean geometry as first theory,
108
,
152

hyperbolic geometry,
139
,
141

neutral geometry,
122
,
131

new axiom system for,
107

non-Euclidean geometries,
8
,
106
,
118
,
121
,
123
,
124–141

projective geometry,
141

revising Euclidean geometry,
51–52

solid geometry,
7

spherical geometry,
125

unity of geometry and arithmetic,
69
,
71
,
91
,
92
,
95
,
110
,
111
,
153–154

Geometry, Euclid and Beyond
(Hartshorne),
47

Glagoleva, E. G.,
99

Gödel, Kurt,
150

Greeks (ancient),
8
,
14
,
15
,
120
,
148

Groups,
142–145
,
151

Grundlagen der Geometrie
(Hilbert),
106
,
107
,
108
,
110
,
111

Guthrie, Francis,
150–151

Hadamard, Jacques,
27

Haken, Wolfgang,
151

Haldane, J. B. S.,
124

Hardy, G. H.,
77

Hartshorne, Robin,
47

Haytham, Ibn al,
120

Hilbert, David,
13
,
27
,
34
,
52
,
80
,
106–115

Homeric epics,
148