The King of Infinite Space (18 page)

Horace,
39

Hyperbola,
99

Hyperbolic plane.
See under
Planes

Hypotenuse,
69
,
100
.
See also
Pythagorean theorem

Identity,
25
,
50–51
,
68
,
142
,
144

of a point and pair of numbers,
112
(
see also
Points: point as pair of numbers
)

between shapes and numbers,
153

Inference,
15
,
17
,
19
,
43
,
44
,
49
,
123

rules of inference,
90

Infinite regress,
32

Infinity,
38
,
49
,
87
,
132
,
134
,
137

natural numbers as potentially infinite,
92–93

Intuition,
22
,
45
,
53
,
54
,
82

Inverse relationship,
81(n)
,
82
,
83
,
105
,
142
,
143
,
144

Isometry,
144

James, Henry,
117

Johnson, Samuel,
33
,
147

Joyce, D. E.,
62

Judt, Tony,
4

Jupiter and Antiope
(painting),
77–78
,
79
,
82
,
87

Kant, Immanuel,
117

Kazan, University of,
128–129

Kazan Messenger, The,
129
,
132

Kirillov, A. A.,
99

Klein, Felix,
140

Kline, Morris,
34

La Géométrie
(Descartes),
96

Latitude/longitude,
3

Leçons de géométrie élémentaire
(Hadamard),
27

Length,
22–23
,
33
,
35
,
36
,
159

Libri Decem
(Vitruvius Pollio),
1

Lines,
79
,
111

curved lines,
137
(
see also
Curvature
)

existence of,
107

hyperbolic lines,
134
,
135

line segments,
95
,
110–111

parallel lines,
34
,
84
,
84(n)
,
88
,
89–90
,
125
,
130
,
138
,
161

straight lines,
7
,
13
,
14
,
22
,
23
,
25
,
33
,
34
,
37
,
38
,
39
,
43
,
46
,
48
,
50
,
52
,
53
,
60
,
61
,
62
,
63
,
66
,
73
,
80–81
,
95
,
98
,
112
,
113
,
135
,
137
,
143
,
159
,
160
,
161

straight lines as ratio of three numbers,
112
,
113

Lobachevsky, Nicolai,
118
,
122–123
,
126
,
128–131
,
133
,
139

Logic,
2
,
12
,
23
,
34
,
53
,
54
,
59
,
65
,
80
,
82
,
83
,
90
,
107
,
108
,
119

of relationships,
24

See also
Syllogisms

Magnitudes,
7
,
94

Mallory, George,
58–59

Mathematical Thought from Ancient to Modern Times
(Kline),
34

Mathematics,
2–3
,
7
,
12
,
41
,
83
,
151

as doubtful,
123

mathematical physics,
144

and mountain-climbing pastoral,
57

Measurements/mensuration,
11

Middle Ages,
80

Mirror images,
68

Models,
13–14
,
108

Modus ponens
,
17
,
82

Moise, Edwin,
94

Monet, Claude,
152

Morality,
58
,
156

Mordell, Louis Joel,
57

Morley, Frank,
147

Motion,
25
,
26
,
27
,
28
,
29
,
63
,
143

as impossible,
43

power of geometrical objects to move or be moved,
36
,
37
,
39
,
52
,
68
,
95
,
145

rigid body moves,
144

ways of moving in a plane,
37

Mountain-climbing pastoral,
57–58

Mount Everest,
58

Multiplication,
103
,
104
,
110
,
112

Newton, Isaac,
47

Non-Euclidean geometries.
See under
Geometry

Nothing,
41
,
42
,
43–44

Notices of the American Mathematical Society
,
150

Numbers,
3
,
7
,
12
,
17
,
29
,
30
,
69
,
145
,
153

and distances,
23

fractions,
38
,
94
,
102
,
103

geometrical properties of numerals,
92

greatest/least numbers,
109

identifying points in space,
36

irrational numbers,
102

natural numbers,
91–92
,
92–93
,
95
,
101

natural numbers as potentially infinite,
92–93

negative numbers,
101–102
,
103
,
142

new numbers,
101–102

number as multitude composed of units,
93

and points,
109
(
see also
Points: point as pair of numbers
)

prime numbers,
100

rational numbers,
94
,
109

real numbers,
94
,
103
,
105–106
,
109
,
110
,
111
,
112

Roman numerals,
4

sets of numbers,
111–112

squaring/square roots of,
70
,
72
,
100
,
101
,
102
,
103
,
110
,
135–136

zero,
101
,
103
,
104
,
143

Oblongs,
161

Omar Khayyám,
120–121

On Nature
(Parmenides),
42

Paintings,
77–79
,
140

Pappus,
68

Papyrus,
8

Parabola,
98

Paradoxes,
38

Parallelism,
53
,
56
,
74
,
74(n)
,
81
,
87
.
See also
Lines: parallel lines
;
Parallel postulate

Parallelograms,
74
,
74(n)
,
75

Parallel postulate,
81(n)
,
117–124

denial/failure of,
118
,
120
,
123
,
131
,
137
,
139–140

and Pythagorean theorem,
119

See also
Axioms: fifth axiom

Parmenides,
42–43
,
44

Parts,
34–35
,
42

whole as greater than the part,
21
,
29–30

Pasch, Moritz,
34

Peirce, C. S.,
23

Perspective (in paintings),
141–142

Peyrard, François,
8

Planes,
14
,
33
,
38
,
39
,
40
,
41
,
94
,
96–97
,
108
,
111
,
112
,
138
,
143
,
144
,
152

defined,
35–36
,
159

degrees of freedom of,
37

existence of,
107

hyperbolic plane,
129–130
,
130(fig.)
,
134
,
135
,
137

projective plane,
141–142

Plato,
5
,
13
,
60
,
95
,
145

Playfair, Francis,
53–54
.
See also
Axioms: Playfair's axiom

Poincaré, Henri,
134

dictionary of,
138–139

Poincaré disk,
134–138
,
135(fig.)

Points,
3
,
7
,
13
,
33
,
37
,
53
,
87
,
111

vs. atoms,
42
,
43

“between two points,”
14
,
41
,
43
,
44
,
46
,
48
,
50
,
61
,
62
,
70
,
95
,
124–125
,
126
,
135
,
137

and continuity,
44

defined,
34
,
35
,
159

existence of,
49
,
107
,
109

hyperbolic points,
134

point as pair of numbers,
97–98
,
100
,
112
,
113–114
,
114–115

Polygons,
48

Postulates,
12
.
See also
Axioms

Praxinoscopes,
78

Precision,
4
,
59

Premises,
15–16
,
90

Principia
(Newton),
47

Proclus,
119

Proofs,
12
,
17
,
19
,
20
,
26
,
31
,
47
,
58
,
59
,
87
,
150

as artifacts,
32

and common beliefs,
20
,
21

as difficult,
65
,
89
,
148

of four-color theorem,
151

by Lobachevsky,
133

of parallel postulate,
119
,
120–122
,
124

proof by contradiction,
83
(
see also
Reductio ad absurdum
)

of Pythagorean theorem,
71–75
,
96

steps in,
59

of twenty-seventh proposition,
83–87
,
90

as way of life,
148
,
156
(
see also
Axiomatic systems: as way of life
)

Proportions,
7
,
94
,
108

Propositions,
6–7
,
11
,
17
,
90

difficulty of,
89

fifth proposition,
58
,
63–68

first proposition,
60–63
,
61(fig.)

first twenty-eight propositions,
122

forty-seventh proposition,
68–75

forty-sixth proposition,
73

fourth proposition,
26–27
,
36
,
39
,
67
,
68
,
74

sixteenth proposition,
83–84
,
84(fig.)
,
84(n)
,
85(fig.)
,
86

third proposition,
66

thirty-second proposition,
119

twenty-ninth proposition,
118
,
155

twenty-seventh proposition,
80–90
,
81(fig.)
,
84(n)
,
86(fig.)

Pseudosphere,
132–133
,
132(fig.)

Ptolemy I,
5

Ptolemy Soter,
58
,
119

Pyramids,
11
,
12

Pythagoreans,
12
,
100

Pythagorean theorem,
68–75
,
72(fig.)
,
100

algebraic equation of,
96

and parallel postulate,
119

Quadrilateral figures,
160
,
161

Railroads,
4
,
141

Ratios,
94
,
100
,
101
,
112–113

Rectangles,
7
,
96

Rectilinear figures,
60
,
160

Reductio ad absurdum
,
77
,
83–87

Reflection (in planes),
37
,
68
,
143
,
144

Relativity and Geometry
(Torretti),
39

Relativity theory,
118

Renaissance,
8
,
141
.
See also
Arab renaissance