The King of Infinite Space (19 page)

Rhombus/rhomboid,
161

Riemann, Bernhard,
118

Rigid objects,
144

Roman empire,
3–4

Roman numerals,
4

Rotation,
37
,
143
,
144

Rubiyat of Omar Khayyám
,
120–121

Rulers.
See
Straight-edge and compass

Ruskin, John,
78

Russell, Bertrand,
7
,
27
,
28
,
123

Saccheri, Girolomo,
121

“Saggio di interpretazione della geometria non-euclidea” (Beltrami),
132

Science,
124

Science and Hypothesis
(Poincaré),
138

Self-evidence,
46
,
55

Shapes,
3
,
7
,
12–13
,
52
,
60
,
71
,
107
,
117
,
133
,
145
,
153

coincidence of,
25–26
,
36
,
39

definition of,
49

Size,
22

Socrates,
15
.
See also
Plato

Some Versions of Pastoral
(Empson),
57–68

Space(s),
4
,
8
,
12
,
20
,
28
,
35
,
37
,
43
,
56
,
70
,
125
,
149

homogeneity of,
52

three-dimensional,
40
,
70
,
144

unbounded vs. infinite,
38

Spheres,
38
,
39
.
See also
Surfaces: surface of a sphere

Square roots.
See
Numbers: squaring/square roots of

Squares,
7
,
72(fig.)
,
75
,
79
,
96
,
161

St. Vincent Millay, Edna,
19

Stability,
31

Steiner-Lehmus theorem,
148

Steinitz, Ernst,
103–104

Straight-edge and compass,
47–48
,
63
,
145

Subtraction,
21
,
24
,
67
,
103
,
104
,
110
,
112

Superposition,
23
,
39
.
See also
Coincidence

Surfaces,
33
,
133
,
159

surface of a sphere,
38
,
40
,
125–126

Syllogisms,
15–16

Theaetetus,
6

Theorema Egregium
(Gauss),
41

Theorems,
25
,
79
,
90
,
147

as being made axioms,
46

forty-first theorem,
74

four-color theorem,
151

of hyperbolic geometry,
139

of Lobachevsky,
131
,
139

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

Steiner-Lehmus theorem,
148

See also
Pythagorean theorem

Theories,
112
,
118

of Euclidean and hyperbolic geometry,
139

Euclidean geometry as first theory,
108
,
152

Thom, René,
95
,
107
,
110
,
149

Time,
4
,
12–13
,
28
,
78
,
87
,
88
,
142
,
149

flow of time vs. points used to mark,
44

and twenty-seventh proposition,
81–82

Torretti, Roberto,
39

Transformations,
143–144
,
145

Translation (in planes),
37
,
143
,
144

Triangles,
7
,
13
,
25
,
28
,
39
,
79
,
84
,
119

and Beltrami pseudosphere,
133

curvilinear,
139

defined,
34
,
160
,
161

as equal,
26
,
67

equilateral,
60–63
,
147
,
160

hyperbolic,
130
,
133(fig.)
,
139

isosceles,
58
,
64–68
,
148
,
160

Platonic,
60
(
see also
Forms
)

right triangle,
68
(
see also
Pythagorean theorem
)

scalene,
160

in spherical geometry,
125

Trilateral figures,
160
,
161

Truth,
16
,
25
,
117
,
127
,
139

Turner, J. M. W.,
152

“
Uber den Zahlbegriff
” (Hilbert),
107

Unity/diversity of experience,
11
,
154

Vector spaces,
114

“Vergleichende Betrachtungen über neuere geometrische Forschungen” (Klein),
140

Vermeer, Johannes,
79

View of Delft
(painting),
79

Vitruvius Pollio, Marcus,
1–2

Void,
42
,
44

Voltaire,
57

Watteau, Antoine,
77
,
79
,
82
,
87

Weyl, Hermann,
44

“Whither Mathematics?” (Davies),
151

Whymper, Edward,
57

Wittgenstein, Ludwig,
155–156

Zeno the Eleatic,
38

Zero.
See under
Numbers