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| Wolff P. — Breakthroughs in mathematics |
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| Ïðåäìåòíûé óêàçàòåëü |
Absurd, reduction to the 60—62 155
Addition 150
Addition in Peano’s theory 166
Algebra, Boole’s symbolism and 271—272 (see also “Boole George”)
Algebra, geometrical problems solved by see “Geometry (Descartes)”
Analytic geometry 96 104
Analytic geometry, advantages of 109—111
Angles in Euclid’s Elements 44—45
Angles in Lobachevski 74—77 84—92
Angles, defined 16 46—47
Angles, measured by Gauss 95
Angles, propositions 26—31 37
Apollonius 96
Archimedes, life of 129
Archimedes, The Sand Reckoner 116—126 129—137
Aristarchus of Samos 117—118 131
Arithmetic, geometry and 103—104
Arithmetic, Peano’s analysis of 164—169 179 188
Arithmetic, postulates in 126—127
Arithmetic, reduced to logic 188—189 190 194 Numbers”)
Bernoulli, Daniel 207
Bernoulli, Jean 207
Bolyai, John 80
Boole, George, accomplishment of 275
Boole, George, background of 268
Boole, George, The Laws of Thought 243—266 268—275
Boole, George, The Laws of Thought, classes of signs 246—256 269
Boole, George, The Laws of Thought, derivation of laws of symbols 256—266 275
Boole, George, The Laws of Thought, laws of symbols 247—256 269—271
Buffon, Comte Georges de 229
Cantor, Georg 140
Chiliagon, in Archimedes 118 133
Circles in Euclid 15—17 (see also “Great circles”)
Classes in Boolean algebra, law of contradiction 265—266 274
Classes in Boolean algebra, symbols for 247—256 264—265 269—271
Classes in definition of number 169—176 184—188
Classes, hereditary 177 180 191
collections see “Classes”
Coloring of maps 217
Common notions in Euclid’s Elements 18 50—51
Conic sections, defined 96
Constructions in Descartes 97—102 104—110
Constructions in Euclid’s Elements 52—59
Constructions, compared with Descartes 106—107
Constructions, propositions 17—21 26—28 37
Continuity and Irrational Numbers (Dedekind) 138—149 152—160
Contradiction, principles of 265—266 274
Converse domain 173
Counting, Archimedes on 116—126 129—137
Counting, decimal system and 137
Counting, Russell on 186 (see also “Arithmetic; Numbers”)
Curves, equations for 107—109
Cutting and not-cutting lines 70 84
Decimal system, Archimedes’ substitute for 137
Dedekind, Continuity and Irrational Numbers 138—149 152—160
Dedekind, Richard, background of 152
Deductiveness as characteristic of mathematics 267—268
Definitions in Euclid’s Elements 16—17 45—47
Definitions in Euclid’s Elements, of numbers 114—115 126—127 150
Descartes, Geometry 97—102
Descartes, Geometry vs. Euclid’s method 104—110
Descartes, Rene, background of 102—103
Division by zero 151—152
Division, number system and 150—151
Domain, defined 173
Duality, law of, in Boole 266
Earth, diameter of, in Archimedes 117 132
Elements of Geometry see “Euclid’s Elements of Geometry”
Euclid, his method vs. Descartes’ 109—110
Euclid, significance of 43
Euclid’s Elements of Geometry —Book I 15—41 43—62
Euclid’s Elements of Geometry —Book I, common notions 18 50—51
Euclid’s Elements of Geometry —Book I, definitions 16—17 45—47
Euclid’s Elements of Geometry —Book I, postulates 17 47—50
Euclid’s Elements of Geometry —Book I, Proposition 1 17—19 52—54
Euclid’s Elements of Geometry —Book I, Proposition 10 26
Euclid’s Elements of Geometry —Book I, Proposition 11 27
Euclid’s Elements of Geometry —Book I, Proposition 12 28
Euclid’s Elements of Geometry —Book I, Proposition 13 29
Euclid’s Elements of Geometry —Book I, Proposition 14 29—30
Euclid’s Elements of Geometry —Book I, Proposition 15 30—31
Euclid’s Elements of Geometry —Book I, Proposition 16 31—32 91—92 94
Euclid’s Elements of Geometry —Book I, Proposition 17 32
Euclid’s Elements of Geometry —Book I, Proposition 18 33
Euclid’s Elements of Geometry —Book I, Proposition 19 33—34
Euclid’s Elements of Geometry —Book I, Proposition 2 19—20 54—57
Euclid’s Elements of Geometry —Book I, Proposition 20 34
Euclid’s Elements of Geometry —Book I, Proposition 21 35
Euclid’s Elements of Geometry —Book I, Proposition 22 36
Euclid’s Elements of Geometry —Book I, Proposition 23 37
Euclid’s Elements of Geometry —Book I, Proposition 24 37—38
Euclid’s Elements of Geometry —Book I, Proposition 25 38—39
Euclid’s Elements of Geometry —Book I, Proposition 26 39—41
Euclid’s Elements of Geometry —Book I, Proposition 27 63—64 81—82
Euclid’s Elements of Geometry —Book I, Proposition 28 64—65
Euclid’s Elements of Geometry —Book I, Proposition 29 65—66 82—83
Euclid’s Elements of Geometry —Book I, Proposition 3 20—21 57
Euclid’s Elements of Geometry —Book I, Proposition 30 66—67
Euclid’s Elements of Geometry —Book I, Proposition 31 67
Euclid’s Elements of Geometry —Book I, Proposition 32 67—68 85
Euclid’s Elements of Geometry —Book I, Proposition 4 21—22 57—59
Euclid’s Elements of Geometry —Book I, Proposition 47 108—109
Euclid’s Elements of Geometry —Book I, Proposition 5 22—23 59—60
Euclid’s Elements of Geometry —Book I, Proposition 6 23—24 60—62
Euclid’s Elements of Geometry —Book I, Proposition 7 24
Euclid’s Elements of Geometry —Book I, Proposition 8 25
Euclid’s Elements of Geometry —Book I, Proposition 9 26
Euclid’s Elements of Geometry —Book II 15
Euclid’s Elements of Geometry —Book III 15
Euclid’s Elements of Geometry —Book IV 15
Euclid’s Elements of Geometry —Book IX 15 103
Euclid’s Elements of Geometry —Book IX Proposition 20 115—116 127—129
Euclid’s Elements of Geometry —Book V 15 43
Euclid’s Elements of Geometry —Book VI 15
Euclid’s Elements of Geometry —Book VII 15 103
Euclid’s Elements of Geometry —Book VII, definitions 114—115 126—127
Euclid’s Elements of Geometry —Book VIII 15 103
Euclid’s Elements of Geometry —Book X 15 43
Euclid’s Elements of Geometry —Book XI 15
Euclid’s Elements of Geometry —Book XII 15
Euclid’s Elements of Geometry —Book XIII 15 43
Eudoxus 43 118
Euler on geometry of position (topology) 197—206 207—215
Euler, Leonhard, background of 205—207
Extraordinary, defined 228
Fifth Postulate of Euclid 17
Fifth Postulate of Euclid in analytic geometry 109
Fifth Postulate of Euclid, Euclid’s first use of 82—83
Fifth Postulate of Euclid, Saccheri’s critique of 78—80
Fifth Postulate of Euclid, substitutes for 83—84 94
Finitude and mathematical induction 175—182 190—194
Fractions 151
Frege, Gottlob 169 180n 181 188
Gambler’s fallacy 238—239
Gauss, Karl Friedrich 152
Gauss, measures angles 95
Gelon, King of Syracuse 116 129 130
Geodesics 93—94
Geometry (Descartes) 97—102
Geometry (Descartes) vs. Euclid’s method 104—110
Geometry, analytic 96 104
Geometry, analytic, advantages of 109—111
Geometry, Euclidean see “Euclid’s Elements of Geometry”
Geometry, non-Euclidean 80
Geometry, non-Euclidean, Riemannian 93 94
Geometry, non-Euclidean, which is most suitable? 94—95 (see also “Lobachevski Nicholas”)
Geometry, of position see “Topology”
Geometry, postulates in 52—53
Geometry, proof in 43
Geometry, proof in reduction to the absurd 60—62
| Geometry, proof in superimposition 58
Geometry, proof in synthetic vs. analytic method 109—110
Geometry, thinking scientifically about 40—43
Great circles, non-Euclidean geometry and 91—94
Heine, E. 139
Heliocentric theory 117—118 130—132
Hereditary property of numbers 177 180 191
Induction, mathematical 165 177—182 190—194
Inductive numbers 182
Inductive property of numbers 177 191
Infinity of numbers in Archimedes 116—126 129—137
Infinity of numbers in Russell 179
Introduction to Mathematical Philosophy (Russell) 161—194
Introduction to Mathematical Philosophy (Russell), definition of number 169—176 184—188
Introduction to Mathematical Philosophy (Russell), finitude and mathematical induction 175—182 190—194
Introduction to Mathematical Philosophy (Russell), series of natural numbers 161—169
Irrational numbers, described by Dedekind’s cut 145—149 152—160
Irrational numbers, discovered 149
Konigsberg, problem of 198—206 207—216
Lagrange, Joseph Louis 219
Language see “Logic”
Laplace, Pierre Simon de, background of 230—231
Laplace, Pierre Simon de, on probability 219—230 233—241
Laws of Thought, The (Boole) 243—266 268—275
Laws of Thought, The (Boole), classes of signs 246—256 269
Laws of Thought, The (Boole), derivation of laws of symbols 256—266 275
Laws of Thought, The (Boole), laws of symbols 247—256 269—271
Leibniz, Baron Gottfried von 197—198 215 220
Lines, cutting and not-cutting 70 84
Lines, Euclid’s definition of 16 45—46
Lines, parallel see “Parallel lines”
Lines, skew 47
Lines, straight see “Straight lines”
Lobachevski, Nicholas, background of 80
Lobachevski, The Theory of Parallels 68—77
Lobachevski, The Theory of Parallels, analytic geometry and 109
Lobachevski, The Theory of Parallels, angles and triangles 74—77 84—92
Lobachevski, The Theory of Parallels, publication of 80
Lobachevski, The Theory of Parallels, substitutes postulate for Euclid’s Fifth 84
Logic, laws of symbols in Boole 247—256 269—271
Logic, laws of symbols in Boole, derivation of 256—266 275
Logic, mathematics and, in Boole 242 267—268
Logic, mathematics and, in Frege 188
Logic, mathematics and, in Peano 188
Logic, mathematics and, in Russell 188—189 190 194
Logic, symbolic 268 275
Maps, coloring of 217
Marcellus 129
Mathematical induction 165 177—182 190—194
Mathematics, common characteristic of 267
Mecanique Celeste 230—231
Mechanical brains 242
Mersenne, Marin 102
Moebius strip 215—217
Multiplication 150
Non-Euclidean geometry 80
Non-Euclidean geometry of Lobachevski 68—77
Non-Euclidean geometry of Lobachevski, analytic geometry and 109
Non-Euclidean geometry of Lobachevski, angles and triangles 74—77 84—92
Non-Euclidean geometry of Lobachevski, postulate substituted for Euclid’s Fifth 84
Non-Euclidean geometry of Lobachevski, published 80
Non-Euclidean geometry, Riemannian 93 94
Non-Euclidean geometry, which is most suitable? 94—95
Numbers, denned by Euclid 114—115 150
Numbers, denned by Russell 169—176 184—188
Numbers, inductive 182
Numbers, irrational, described by Dedekind’s cut 145—149 152—160
Numbers, irrational, discovered 149
Numbers, natural 150
Numbers, natural, theory of 164—169 176—182 190—194
Numbers, quantity of, in Archimedes 116—126 129—130
Numbers, quantity of, in Euclid 115—116
Numbers, quantity of, in Russell 175—182 190—194
Numbers, rational, compared with points on straight line 141—145 152—158
Numbers, rational, defined 152
Numbers, rational, properties of 140—142
Opposite, reduction to the 128—129
Parallel lines in Euclid’s Elements 44
Parallel lines in Euclid’s Elements, defined 17 47
Parallel lines in Euclid’s Elements, propositions 63—68 80—83
Parallel lines in Lobachevski 84 91—93
Parallel lines in Lobachevski, theorems 70—74
Parallel lines, three possibilities for 94
Parallelograms in Euclid 44
Peano, Giuseppe 183
Peano, number theory of 164—169 179 188
Pheidias 118
Planetary theory in Aristarchus and Archimedes 117—118 131
Planetary theory, Laplace’s work on 230—231
Poincare, Henri 181
Point, defined in Euclid’s Elements 16 45—46
Position, relative see “Topology”
Posterity of numbers 177—178 180 192—193
Postulates in Descartes 109
Postulates in Euclid’s Elements 17 47—50
Postulates in Euclid’s Elements, lacking in number theory 126—127 (see also “Fifth Postulate of Euclid”)
Postulates, function of, in geometry 52—53
Prime numbers in Euclid, defined 114
Prime numbers in Euclid, quantity of 114—116 127—129
probability 218—241
Probability, a priori vs. a posteriori arguments 233
Probability, defined 224 234 235
Probability, gambler’s fallacy in 238—239
Probability, Laplace on 219—230 233—241
Probability, rules for 224—230 234—241
Probability, truth and 231
Progressions, Peano’s theory and 167
Proofs, , reduction to the opposite 128—129
Proofs, method of superimposition 58
Proofs, reduction to the absurd 60—62 155
Proofs, synthetic vs. analytic method 109—110
Propositions of Euclid see “Euclid’s Elements of Geometry”
Pseudosphere 93—94
Pythagoras 149 164
Pythagorean theorem of Euclid 108—109
Q.E.F. 53
Rational numbers, compared with points on straight line 141—145 152—158
Rational numbers, defined 152
Rational numbers, properties of 140—142
Reason, derivation of symbols of logic from laws of 256—266 275
Reasonable degree of belief 220 232
Reduction in Russell’s definition of number 188
Reduction to the absurd 60—62 155
Reduction to the opposite 128—129
Relations, signs of, in Boole 252—256
Relations, types of, in Russell 173
Relative position see “Topology”
Riemann, Bernhard 80
Riemann, geometry of 93 94
Right angles, Euclid’s definition of 16 46—47
Russell, Bertrand, background of 183—184
Russell, finitude and mathematical induction 175—182 190—194
Russell, finitude and mathematical induction, series of rational numbers 161—169
Russell, Introduction to Mathematical Philosophy 161—194
Russell, Introduction to Mathematical Philosophy, definition of number 169—176 184—188
Saccheri, Girolamo, on Euclid’s Fifth Postulate 78—80
Sand Reckoner, The (Archimedes) 116—126 129—137
Schnitt, Dedekind’s 145—149 152—160
Science, object of 256
Sets see “Classes”
Signs, classes of 246—156 269
Signs, defined 244 (see also “Laws of Thought The”)
Similarity and class 174 185 188
Skew lines 47
Sphere, non-Euclidean geometry and 91—94
Square roots see “Irrational numbers”
Stadium (Greekmeasure), length of 132
Straight lines in Euclid’s Elements 44 52—57
Straight lines in Euclid’s Elements, defined 16 45—46
Straight lines in Euclid’s Elements, propositions 17—21 26—28
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