One Hundred Great Problems of Elementary Mathematics: Their History and Solution(Dover Books on Mathema 1.4tics)

Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, etc. Features squaring the circle, pi, similar problems. No advanced math is required. Includes 100 problems with proofs. "The collection, drawn from arithmetic, algebra, pure and algebraic geometry and astronomy, is extraordinarily interesting and attractive." - Mathematical Gazette This uncommonly interesting volume covers 100 of the most famous historical problems of elementary mathematics. Not only does the book bear witness to the extraordinary ingenuity of some of the greatest mathematical minds of history - Archimedes, Isaac Newton, Leonhard Euler, Augustin Cauchy, Pierre Fermat, Carl Friedrich Gauss, Gaspard Monge, Jakob Steiner, and many others - but it provides rare insight and inspiration to any reader, from high school math student to professional mathematician. This is indeed an unusual and uniquely valuable book. The one hundred problems are presented in six categories: 26 arithmetical problems, 15 planimetric problems, 25 classic problems concerning conic sections and cycloids, 10 stereometric problems, 12 nautical and astronomical problems, and 12 maxima and minima problems. In addition to defining the problems and giving full solutions and proofs, the author recounts their origins and history and discusses personalities associated with them. Often he gives not the original solution, but one or two simpler or more interesting demonstrations. In only two or three instances does the solution assume anything more than a knowledge of theorems of elementary mathematics; hence, this is a book with an extremely wide appeal. Some of the most celebrated and intriguing items are: Archimedes' "Problema Bovinum," Euler's problem of polygon division, Omar Khayyam's binomial expansion, the Euler number, Newton's exponential series, the sine and cosine series, Mercator's logarithmic series, the Fermat-Euler prime number theorem, the Feuerbach circle, the tangency problem of Apollonius, Archimedes' determination of pi, Pascal's hexagon theorem, Desargues' involution theorem, the five regular solids, the Mercator projection, the Kepler equation, determination of the position of a ship at sea, Lambert's comet problem, and Steiner's ellipse, circle, and sphere problems. This translation, prepared especially for Dover by David Antin, brings Dörrie's "Triumph der Mathematik" to the English-language audience for the first time.

Table of Contents:
ARITHMETICAL PROBLEMS 1. Archimedes' Problem Bovinum 2. The Weight Problem of Bachet de Meziriac 3. Newton's Problem of the Fields and Cows 4. Berwick's Problem of the Seven Sevens 5. Kirkman's Schoolgirl Problem 6. The Bernoulli-Euler Problem of the Misaddressed Letters 7. Euler's Problem of Polygon Division 8. Lucas' Problem of the Married Couples 9. Omar Khayyam's Binomial Expansion 10. Cauchy's Mean Theorem 11. Bernoulli's Per Sum Problem 12. The Euler Number 13. Newton's Exponential Series 14. Nicolaus Macerator's Logarithmic Series 15. Newton's Sine and Cosine Series 16. Andre's Derivation of the Secant and Tangent Series 17. Gregory's Arc Tangent Series 18. Buffon's Needle Problem 19. The Fermat-Euler Prime Number Theorem 20. The Fermat Equation 21. The Fermat-Gauss Impossibility Theorem 22. The Quadratic Reciprocity Law 23. Gauss' Fundamental Theorem of Algebra 24. Sturm's Problem of the Number of Roots 25. Abel's Impossibility Theorem 26. The Hermite-Lindemann Transcendence Theorem PLANIMETRIC PROBLEMS 27. Euler's Straight Line 28. The Feuerbach Circle 29. Castillon's Problem 30. Malfatti's Problem 31. Monge's Problem 32. The Tangency Problem of Apollonius 33. Mascheroni's Compass Problem 34. Steiner's Straight-edge Problem 35. The Delian Cube-doubling Problem 36. Trisection of an Angle 37. The Regular Heptadecagon 38. Archimedes' Determination of the Number Pi 39. Fuss' Problem of the Chord-Tangent Quadrilateral 40. Annex to a Survey 41. Alhazen's Billiard Problem PROBLEMS CONCERNING CONIC SECTIONS AND CYCLOIDS 42. An Ellipse from Conjugate Radii 43. An Ellipse in a Parallelogram 44. A Parabola from Four Tangents 45. A Parabola from Four Points 46. A Hyperbola from Four Points 47. Van Schooten's Locus Problem 48. Cardan's Spur Wheel Problem 49. Newton's Ellipse Problem 50. The Poncelet-Brianchon Hyperbola Problem 51. A Parabola as Envelope 52. The Astroid 53. Steiner's Three-pointed Hypocycloid 54. The Most Nearly Circular Ellipse Circumscribing a Quadrilateral 55. The Curvature of Conic Sections 56. Archimedes' Squaring of a Parabola 57. Squaring a Hyperbola 58. Rectification of a Parabola 59. Desargue's Homology Theorem (Theorem of Homologous Triangles) 60. Steiner's Double Element Construction 61. Pascal's Hexagon Theorem 62. Brianchon's Hexagram Theorem 63. Desargues' Involution Theorem 64. A Conic Section from Five Elements 65. A Conic Section and a Straight Line 66. A Conic Section and a Point STEREOMETRIC PROBLEMS 67. Steiner's Division of Space by Planes 68. Euler's Tetrahedron Problem 69. The Shortest Distance Between Skew Lines 70. The Sphere Circumscribing a Tetrahedron 71. The Five Regular Solids 72. The Square as an Image of a Quadrilateral 73. The Pohlke-Schwartz Theorem 74. Gauss' Fundamental Theorem of Axonometry 75. Hipparchus' Stereographic Projection 76. The Mercator Projection NAUTICAL AND ASTRONOMICAL PROBLEMS 77. The Problem of the Loxodrome 78. Determining the Position of a Ship at Sea 79. Gauss' Two-Altitude Problem 80. Gauss' Three-Altitude Problem 81. The Kepler Equation 82. Star Setting 83. The Problem of the Sundial 84. The Shadow Curve 85. Solar and Lunar Eclipses 86. Sidereal and Synodic Revolution Periods 87. Progressive and Retrograde Motion of the Planets 88. Lambert's Comet Problem EXTREMES 89. Steiner's Problem Concerning the Euler Number 90. Fagnano's Altitude Base Point Problem 91. Fermat's Problem for Torricelli 92. Tacking Under a Headwind 93. The Honeybee Cell (Problem by Reaumur) 94. Regiomontanus' Maximum Problem 95. The Maximum Brightness of Venus 96. A Comet Inside the Earth's Orbit 97. The Problem of the Shortest Twilight 98. Steiner's Ellipse Problem 99. Steiner's Circle Problem 100. Steiner's Sphere Problem Index of Names