TABLE OF CONTENTS: Preface v
Introduction 5
1. The Sequence of Prime Numbers 9
2. Traversing Nets of Curves 13
3. Some Maximum Problems 17
4. Incommensurable Segments and Irrational Numbers 22
5. A Minimum Property of the Pedal Triangle 27
6. A Second Proof of the Same Minimum Property 30
7. The Theory of Sets 34
8. Some Combinatorial Problems 43
9. On Waring's Problem 52
10. On Closed Self-Intersecting Curves 61
11. Is the Factorization of a Number into Prime Factors Unique?66
12. The Four-Color Problem 73
13. The Regular Polyhedrons 82
14. Pythagorean Numbers and Fermat's Theorem 88
15. The Theorem of the Arithmetic and Geometric Means 95
16. The Spanning Circle of a Finite Set of Points 103
17. Approximating Irrational Numbers by Means of Rational Numbers ill
18. Producing Rectilinear Motion by Means of Linkages 119
19. Perfect Numbers 129
20. Euler's Proof of the Infinitude of the Prime Numbers 135
21. Fundamental Principles of Maximum Problems 139
22. The Figure of Greatest Area with a Given Perimeter 142
23. Periodic Decimal Fractions 147
24. A Characteristic Property of the Circle 160
25. Curves of Constant Breadth 163
26. The Indispensability of the Compass for the Constructions of Elementary Geometry 177
27. A Property of the Number 30 187
28. An Improved Inequality 192
Notes and Remarks 197
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