| Preface |
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| Hints to the Reader |
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xi | |
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3 | (9) |
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Examples and Comments on Chapter I, 1-14 |
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The logician, the mathematician, the physicist, and the engineer |
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Generalization, Specialization, Analogy |
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12 | (23) |
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Generalization, specialization, analogy, and induction |
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Generalization, specialization, and analogy |
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Examples and Comments on Chapter II, 1-46 |
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[First Part, 1-20; Second Part, 21-46] |
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A representative special case |
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An objection and a first approach to a proof |
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A second approach to a proof |
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Induction in Solid Geometry |
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35 | (24) |
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First supporting contacts |
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Verifications and verifications |
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Generalization, specialization, analogy |
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An array of analogous problems |
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Many problems may be easier than just one |
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Prediction and verification |
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Induction suggests deduction, the particular case suggests the general proof |
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Examples and Comments on Chapter III, 1-41 |
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Induction: adaptation of the mind, adaptation of the language |
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Descartes' work on polyhedra |
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Supplementary solid angles, supplementary spherical polygons |
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Induction in the Theory of Numbers |
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59 | (17) |
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Right triangles in integers |
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On the sum of four odd squares |
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Tabulating the observations |
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On the nature of inductive discovery |
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On the nature of inductive evidence |
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Examples and Comments on Chapter IV, 1-26 |
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Miscellaneous Examples of Induction |
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76 | (14) |
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The role of the inductive phase |
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Examples and Comments on Chapter V, 1-18 |
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Explain the observed regularities |
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Classify the observed facts |
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90 | (18) |
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Transition to a more general viewpoint |
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Schematic outline of Euler's memoir |
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Examples and Comments on Chapter VI, 1-25 |
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A combinatorial problem in plane geometry |
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Another recursion formula |
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Another Most Extraordinary Law of the Numbers concerning the Sum of their Divisors |
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How Euler missed a discovery |
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A generalization of Euler's theorem on σ(n) |
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108 | (13) |
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The technique of mathematical induction |
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Examples and Comments on Chapter VII, 1-18 |
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To prove more may be less trouble |
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121 | (21) |
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The pattern of the tangent level line |
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The pattern of partial variation |
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The theorem of the arithmetic and geometric means and its first consequences |
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Examples and Comments on Chapter VIII, 1-63; [First Part, 1-32; Second Part, 33-63] |
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Minimum and maximum distances in plane geometry |
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Minimum and maximum distances in solid geometry |
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The principle of the crossing level line |
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The principle of partial variation |
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Existence of the extremum |
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A modification of the pattern of partial variation: An infinite process |
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Another modification of the pattern of partial variation: A finite process |
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Right prism with square base |
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Right double pyramid with square base |
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General right double pyramid |
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Applying geometry to algebra |
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Applying algebra to geometry |
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Right pyramid with square base |
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142 | (26) |
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Mechanical interpretation |
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Jean Bernoulli's discovery of the brachistochrone |
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Archimedes' discovery of the integral calculus |
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Examples and Comments on Chapter IX, 1-38 |
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Triangle with minimum perimeter inscribed in a given triangle |
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Traffic center of four points in space |
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Traffic center of four points in a plane |
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Traffic network for four points |
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Shortest lines on a polyhedral surface |
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Shortest lines (geodesics) on a curved surface |
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A construction by paper-folding |
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The Calculus of Variations |
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From the equilibrium of cross-sections to the equilibrium of the solids |
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Archimedes' Method in retrospect |
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The Isoperimetric Problem |
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168 | (22) |
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Descartes' inductive reasons |
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Lord Rayleigh's inductive reasons |
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Three forms of the Isoperimetric Theorem |
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Applications and questions |
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Examples and Comments on Chapter X, 1-43; [First Part, 1-15; Second Part, 16-43] |
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Could you derive some part of the result differently? |
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Can you use the method for some other problem? |
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Sharper form of the Isoperimetric Theorem |
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Two sticks and two strings |
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Dido's problem in solid geometry |
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Bisectors of a plane region |
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Bisectors of a closed surface |
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A figure of many perfections |
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Further Kinds of Plausible Reasons |
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190 | (20) |
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Conjectures and conjectures |
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Judging by a related case |
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Judging by the general case |
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Preferring the simpler conjecture |
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Usual heuristic assumptions |
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Examples and Comments on Chapter XI, 1-23 |
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Some usual heuristic assumptions |
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Numerical computation and the engineer |
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| Final remark |
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210 | (3) |
| Solutions to problems |
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213 | (66) |
| Bibliography |
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279 | |
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