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| Ïîèñê ïî óêàçàòåëÿì |
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| Peter Wolff — 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—72 (see also Boole George)
Algebra, geometrical problems solved by (see Geometry (Descartes))
Analytic geometry 96 104
Angles in Euclid’s Elements 44—45
Angles in Euclid’s Elements, defined 16 46—47
Angles in Euclid’s Elements, propositions 26—31 37
Angles in Lobachevski 74—77 84—92
Angles, measured by Gauss 95
Apollonius 96
Archimedes, life of 129
Archimedes, The Sand Reckoner 116—26 129—37
Aristarchus of Samos 117—18 131
Arithmetic, geometry and 103—4
Arithmetic, Peano’s analysis of 164—69 179 188
Arithmetic, postulates in 126—27
Arithmetic, reduced to logic 188—89 190 194 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—66 268—75
Boole, George, The Laws of Thought, classes of signs 246—56 269
Boole, George, The Laws of Thought, derivation of laws of symbols 256—66 275
Boole, George, The Laws of Thought, laws of symbols 247—56 269—71
Buff on, 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 in definition of number 169—76 184—88
Classes in Boolean algebra in definition of number Book II 15
Classes in Boolean algebra in definition of number Book III 15
Classes in Boolean algebra in definition of number Book IV 15
Classes in Boolean algebra in definition of number Book IX 15 103
Classes in Boolean algebra in definition of number Book IX, Proposition 20 115—16 127—29
Classes in Boolean algebra in definition of number Book V 15 43
Classes in Boolean algebra in definition of number Book VI 15
Classes in Boolean algebra in definition of number Book VII 15 103
Classes in Boolean algebra in definition of number Book VII, definitions 114—15 126—27
Classes in Boolean algebra in definition of number Book VIII 15 103
Classes in Boolean algebra in definition of number Book X 15 43
Classes in Boolean algebra in definition of number Book XI 15
Classes in Boolean algebra in definition of number Book XII 15
Classes in Boolean algebra in definition of number Book XIII 15 43
Classes in Boolean algebra in definition of number Proposition 23 37
Classes in Boolean algebra in definition of number Proposition 24 37—38
Classes in Boolean algebra in definition of number Proposition 25 38—39
Classes in Boolean algebra in definition of number Proposition 26 39—41
Classes in Boolean algebra in definition of number Proposition 27 63—64 81—82
Classes in Boolean algebra in definition of number Proposition 28 64—65
Classes in Boolean algebra in definition of number Proposition 29 65—66 82—83
Classes in Boolean algebra in definition of number Proposition 30 66—67
Classes in Boolean algebra in definition of number Proposition 31 67
Classes in Boolean algebra in definition of number Proposition 32 67—68 85
Classes in Boolean algebra in definition of number Proposition 47 108—9
Classes in Boolean algebra, law of contradiction 265—66 274
Classes in Boolean algebra, symbols for 247—56 264—65 269—71
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—10
Constructions in Euclid’s Elements 52—59
Constructions in Euclid’s Elements, compared with Descartes 106—7
Constructions in Euclid’s Elements, propositions 17—21 26—28 37
Continuity and Irrational, Numbers (Dedekind) 138—49 152—60
Contradiction, principles of 265—66 274
Converse domain 173
Counting, Archimedes on 116—26 129—37
Counting, decimal system and 137
Counting, Russell on 186 (see also Arithmetic Numbers)
Curves, equations for 107—9
Cutting and not-cutting lines 70 84
Decimal system, Archimedes’ substitute for 137
Dedekind, Richard, background of 152
Dedekind, Richard, Continuity and Irrational Numbers 138—49 152—60
Deductiveness as characteristic of mathematics 267—68
Definitions in Euclid’s Elements 16—17 45—47
Definitions in Euclid’s Elements of numbers 114—15 126—27 150
Descartes, Rene, background of 102—3
Descartes, Rene, Geometry 97—102
Descartes, Rene, Geometry vs. Euclid’s method 104—10
Division by zero 151—52
Division, number system and 150—51
Domain, defined 173
Duality, law of, in Boole 266
Earth, diameter of, in Archimedes 117 132
Elements of Geometry (see Euclid’s)
Euclid, his method vs. Descartes’ 109—10
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 3 20—21 57
Euclid’s Elements of Geometry - Book I, Proposition 4 21—22 57—59
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
Eudoxus 43 118
Euler, Leonhard on geometry of position (topology) 197—206 207—15
Euler, Leonhard, background of 205—7
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—82 190—94
Fractions 151
Frege, Gottlob 169 180 1 181 188
Gambler’s fallacy 238—39
Gauss, Karl Friedrich 152
Gauss, Karl Friedrich, measures angles 95
Gelon, King of Syracuse 116 129 130
Geodesics 93—94
Geometry (Descartes) 97—102
Geometry (Descartes) vs. Euclid’s method 104—10
Geometry of position (see Topology)
Geometry, analytic 96 104
Geometry, analytic advantages of 109—11
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, 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—10
| Geometry, thinking scientifically about 40—43
Great circles, non-Euclidean, geometry and 91—94
Heine, E. 139
Heliocentric theory 117—18 130—32
Hereditary property of numbers 177 180 191
Induction, mathematical 165 177—82 190—94
Inductive numbers 182
Inductive property of numbers 177 191
Infinity of numbers in Archimedes 116—26 129—37
Infinity of numbers in Russell 179
Introduction to Mathematical, definition of number 169—76 184—88
Introduction to Mathematical, finitude and mathematical induction 175—82 190—94
Introduction to Mathematical, Philosophy (Russell) 161—94
Introduction to Mathematical, series of natural numbers 161—69
Irrational numbers, described by Dedekind’s cut 145—49 152—60
Irrational numbers, discovered 149
Konigsberg, problem of 198—206 207—16
Lagrange, Joseph Louis 219
Language (see Logic)
Laplace, Pierre Simon de background of 230—31
Laplace, Pierre Simon on probability 219—30 233—41
Laws of Thought, classes of signs 246—56 269
Laws of Thought, derivation of laws of symbols 256—66 275
Laws of Thought, laws of symbols 247—56 269—71
Laws of Thought, The (Boole) 243—66 268—75
Leibniz, Baron Gottfried von 197—98 215 220
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, analytic geometry and 109
Lobachevski, Nicholas, angles and triangles 74—77 84—92
Lobachevski, Nicholas, background of 80
Lobachevski, Nicholas, publication of 80
Lobachevski, Nicholas, substitutes postulate for Euclid’s Fifth 84
Lobachevski, Nicholas, The Theory of Parallels 68—77
Logic, derivation of 256—66 275
Logic, laws of symbols in Boole 247—56 269—71
Logic, mathematics and symbolic 268 275
Logic, mathematics in Boole 242 267—68
Logic, mathematics in Frege 188
Logic, mathematics in Peano 188
Logic, mathematics in Russell 188—89 190 194
Maps, coloring of 217
Marcellus 129
Mathematical induction 165 177—82 190—94
Mathematics, common, characteristic of 267
Mecanique Celeste 230—31
Mechanical brains 242
Mersenne, Marin 102
Moebius strip 215—17
Multiplication 150 128—29
Multiplication synthetic vs. analytic method 109—10
Non — Euclidean geometry 80
Non — Euclidean geometry of Lobachevski 68—77
Non — Euclidean geometry which is most suitable 94—95
Non — Euclidean geometry, analytic geometry and 109
Non — Euclidean geometry, angles and triangles 74—77 84—92
Non — Euclidean geometry, postulate substituted for Euclid’s Fifth 84
Non — Euclidean geometry, published 80
Non — Euclidean geometry, Riemannian 93 94
Numbers, defined by Euclid 114—15 150
Numbers, defined by Russell 169—76 184—88
Numbers, inductive 182
Numbers, irrational described by Dedekind’s cut 145—49 152—60
Numbers, irrational discovered 149
Numbers, natural 150
Numbers, natural, theory of 164—69 176—82 190—94
Numbers, quantity of, in Archimedes 116—26 129—30
Numbers, quantity of, in Euclid 115—16
Numbers, quantity of, in Russell 175—82 190—94
Numbers, rational, compared with points on straight line 141—45 152—58
Numbers, rational, defined 152
Numbers, rational, properties of 140—42
Opposite, reduction to the 128—29
Parallel lines in Euclid’s Elements 44
Parallel lines in Lobachevski 84 91—93
Parallel lines in Lobachevski, theorems 70—74
Parallel lines, defined 17 47
Parallel lines, propositions 63—68 80—83
Parallel lines, three possibilities for 94
Parallelograms in Euclid 44
Peano, Giuseppe 183
Peano, Giuseppe, number theory of 164—69 179 188
Pheidias 118
Planetary theory in Aristarchus and Archimedes 117—18 131
Planetary theory, Laplace’s work on 230—31
Poincare, Henri 181
Point, defined in Euclid’s Elements 16 45—46
Position, relative (see Topology)
Posterity of numbers 177—78 180 192—93
Postulates in Descartes 109
Postulates in Euclid’s Elements 17 47—50
Postulates, function of, in geometry 52—53
Postulates, lacking in number theory 126—27 (see also Fifth Postulate of Euclid)
Prime numbers in Euclid, defined 114
Prime numbers in Euclid, quantity of 114—16 127—29
probability 218—41
Probability a priori vs. a posteriori, arguments 233
Probability, defined 224 234 235
Probability, gambler’s fallacy in 238—39
Probability, Laplace on 219—30 233—41
Probability, rules for 224—30 234—41
Probability, truth and 231
Progressions, Peano’s theory and 167
Proofs, method of superimposition 58
Proofs, reduction to the absurd 60—62 155
Proofs, reduction to the opposite, hereditary 177 180 191
Propositions of Euclid (see Euclid’s Elements of Geometry)
Pseudosphere 93—94
Pythagoras 149 164
Pythagorean theorem of Euclid 108—9
Q.E.F. 53
Rational numbers, compared with points on straight line 141—45 152—58
Rational numbers, defined 152
Rational numbers, properties of 140—42
Reason, derivation of symbols of logic from laws of 256—66 275
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—29
Relations, signs of, in Boole 252—56
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—84
Russell, Bertrand, definition of number 169—76 184—88
Russell, Bertrand, induction 175—82 190—94
Russell, Bertrand, Introduction to Mathematical Philosophy 161—94
Russell, Bertrand, series of rational numbers 161—69
Saccheri, Girolamo, on Euclid’s Fifth Postulate 78—80
Sand Reckoner, The (Archimedes) 116—26 129—37
Schnitt, Dedekind’s 145—49 152—60
Science, object of 256
Sets (see Classes)
Signs, classes of 246—56 269
Signs, defined 244 (see also Laws of Thought)
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 Lobachevski 70 76—77 84 92—94
Straight lines in Riemann 93 94
Straight lines, defined 16 45—46
Straight lines, propositions 17—21 26—28
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