| 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 |
