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
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
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
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
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