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Introduction | 1 | |
Part 1 | Arithmetic | |
I. | Calculating with Natural Numbers | 6 |
1. | Introduction of Numbers in the Schools | 6 |
2. | The Fundamental Laws of Reckoning | 8 |
3. | The Logical Foundations of Operations with Integers | 10 |
4. | Practice in Calculating with Integers | 17 |
II. | The First Extension of the Notion of Number | 22 |
1. | Negative Numbers | 23 |
2. | Fractions | 28 |
3. | Irrational Numbers | 31 |
III. | Concerning Special Properties of Integers | 37 |
IV. | Complex Numbers | 55 |
1. | Ordinary Complex Numbers | 55 |
2. | Higher Complex Numbers, especially Quaternions | 58 |
3. | Quaternion Multiplication--Rotation and Expansion | 65 |
4. | Complex Numbers in School Instruction | 75 |
Concerning the Modern Development and the General Structure of Mathematics | 77 | |
Part 2 | Algebra | |
I. | Real Equations with Real Unknowns | 87 |
1. | Equations with one parameter | 87 |
2. | Equations with two parameters | 88 |
3. | Equations with three parameters [lambda], [mu], [nu] | 94 |
II. | Equations in the field of complex quantities | 101 |
A. | The fundamental theorem of algebra | 101 |
B. | Equations with a complex parameter | 104 |
1. | The "pure" equation | 110 |
2. | The dihedral equation | 115 |
3. | The tetrahedral, the octahedral, and the icosahedral equations | 120 |
4. | Continuation: Setting up the Normal Equation | 124 |
5. | Concerning the Solution of the Normal Equations | 130 |
6. | Uniformization of the Normal Irrationalities by Means of Transcendental Functions | 133 |
7. | Solution in Terms of Radicals | 138 |
8. | Reduction of Genral Equations to Normal Equations | 141 |
Part 3 | Analysis | |
I. | Logarithmic and Exponential Functions | 144 |
1. | Systematic Account of Algebraic Analysis | 144 |
2. | The Historical Development of the Theory | 146 |
3. | The Theory of Logarithms in the Schools | 155 |
4. | The Standpoint of Function Theory | 156 |
II. | The Goniometric Functions | 162 |
1. | Theory of the Goniometric Functions | 162 |
2. | Trigonometric Tables | 169 |
A. | Purely Trigonometric Tables | 170 |
B. | Logarithmic--Trigonometric Tables | 172 |
3. | Applications of Goniometric Functions | 175 |
A. | Trigonometry, in particular, spherical trigonometry | 175 |
B. | Theory of small oscillations, especially those of the pendulum | 186 |
C. | Representation of periodic functions by means of series of goniometric functions (trigonometric series) | 190 |
III. | Concerning Infinitesimal Calculus Proper | 207 |
1. | General Considerations in Infinitesimal Calculus | 207 |
2. | Taylors Theorem | 223 |
3. | Historical and Pedagogical Considerations | 234 |
Supplement | ||
I. | Transcendence of the Numbers e and [pi] | 237 |
II. | The Theory of Assemblages | 250 |
1. | The Power of an Assemblage | 251 |
2. | Arrangement of the Elements of an Assemblage | 262 |
Index of Names | 269 | |
Index of Contents | 271 |
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